UC-NRLF 


PRACTICAL 

HYDRAULIC 


FOR  THE 


Distribution  of 
Water  Through  Long  Pipes, 


E.  SHERMAN  GOULD, 


O 

>- 


OS4- 
*> 


REESE    LIBRARY 

OF   THE 

UNIVERSITY   OF  CALIFORNIA. 


Accessions  No.  -4.  3  3.  2-0     Shelf  No. 


PRACTICAL 


HYDRAULIC  FORMULE 


FOR    THE 


Distribution  of 
Water  Through  Long  Pipes. 


E.    SHERMAN    GOULD, 

M.   AM.  SOC.    <  .    K. 

Consult-in (r  Engineer  to  the  Scranton  Gas  and  Water  Co. 


XE\V  YORK  : 
ENGINEERING    NEWS    PUBLISHING    CO. 


C£ 


COPYRIGHTED,  1889, 
ENGINEERING  NEWS  PUBLISHING  CO. 

^  *>  .5  a-a 


INTRODUCTION. 


The  following  pages  first  appeared  as  a  series  of  articles 
in  the  columns  of  ENGINEERING  NEWS.  They  are  now  repub- 
lished  with  a  few  corrections  and  additions. 

In  virtue  of  the  law  of  gravitation,  water  tends  naturally  to 
pass  from  a  higher  to  a  lower  level,  and  without  a  difference  of 
level  there  can  be  no  natural  flow. 

It  can  be  said  in  all  seriousness— although  the  statement  may 
seem  to  invite  the  unjust  accusation  of  an  ill-timed  attempt  at 
pleasantry— that  the  whole  science  of  hydraulics  is  founded  upon 
the  three  following  homely  and  unassailable  axioms : 

First.    That  water  always  seeks  its  own  lowest  level. 

Second.    That,  therefore,  it  always  tends  to  run  down  hill,  and 

Third,  that  other  things  being  equal,  the  steeper  the  hill,  the 
faster  it  runs. 

In  the  case  of  water  flowing  through  long  pipes,  the  hill  down 
which  it  tends  to  run  is  the  HYDRAULIC  GRADE  LINE.  If  the  pipe  be 
of  uniform  diameter  and  character,  the  hydraulic  grade  line  is  a 
straight  line  joining  the  water  surfaces  at  its  two  extremities,  pro- 
vided that  the  pipe  lies  wholly  below  such  straight  line,  and  its  de. 
clivity  is  measured — like  that  of  all  hills — by  the  ratio  of  its  height 
to  its  length. 

But  if  there  be  any  changes  whatever  in  the  pipe,  either  of  dia- 
meter or  in  the  nature  of  its  inside  surface ;  or  if  there  be  increase 
or  diminution  of  the  volume  of  water  entering  it  at  its  upper  ex- 
tremity by  reason  of  branches  leading  to  or  from  the  main  pipe, 
then  the  hydraulic  grade  line  becomes,  broken  and  distorted  to  a 
greater  or  less  extent,  so  that  its  declivity  is  not  uniform  from  end 


INTRODUCTION. 

to  end,  but  consists  of  a  series  of  varying  grades  some  steeper  than 
others  though  all  sloping  in  the  same  direction. 

As  regards  the  third  axiom,  the  proviso—"  other  things  being 
equal" — must  not  be  overlooked.  For  we  shall  find  that  a  pipe  of 
greater  diameter  but  less  hydraulic  declivity  than  another,  may 
give  a  greater  velocity  to  the  water  passing  through  it.  Also,  of 
two  pipes  of  the  same  hydraulic  slope  and  diameter,  the  one  having 
the  smoother  inside  surface  affords  the  greater  velocity. 

The  vertical  distance  from  any  point  in  a  pipe  to  the  hydraulic 
grade  line,  constitutes  the  Piezometric  height,  and  measures  the 
hydraulic  pressure  at  that  point.  It  will  be  seen  that  the  solution 
of  problems  relating  to  the  flow  of  water  through  pipes,  lies  in  the 
knowing  or  ascertaining  of  the  piezometric  height  at  any  desired 
point.  In  general,  it  is  necessary  to  establish  the  piezometric 
height  for  every  point  of  change  of  any  kind  which  occurs  through- 
out the  entire  length  of  the  conduit.  The  joining  of  the  upper  ex- 
tremities of  these  heights  gives  the  complete  hydraulic  grade  line. 

The  object  of  the  following  papers  is  to  establish  systematic 
methods  for  tracing  the  hydraulic  grade  line  under  the  different 
circumstances  likely  to  occur  in  practice,  and  generally,  to  furnish 
solutions  for  a  large  number  of  practical  problems,  commencing 
with  the  simplest  cases  and  extending  to  some  rather  intricate 
ones,  not  usually  embraced  in  our  hydraulic  manuals. 

E.  8.  G. 

SCRANTON,  Pa.,  May,  1889. 


TABLE     OF   CON  TENTS. 

INTRODUCTION. 

CHAPTER  I. 

Flow  through  a  short  horizontal  pipe— Effect  on  velocity  of  increased 
length— Fractional  head  —Hydraulic  grade  line— Hydrostatic  and  hy- 
draulic pressures— Piezometric  tubes— Result  of  raising  a  pipe  line  above 
the  hvdraulic  grade  line— Why  the  water  ceases  to  rise  in  the  upper 
stories  of  the  houses  of  a  town  when  the  consumption  is  increased— In- 
fluence of  inside  surface  of  pipes  upon  velocity  of  flow— Darcy's  coef- 
ficients—Fundamental equations— Length  of  a  pipe  line  usually  deter- 
mined by  its  horizontal  projection— Numerical  examples  of  simple  and 
compound  systems. 

CHAPTER  II. 

Calculations  are  the  same  for  pipes  laid  horizontally  or  on  a  slope- 
Qualification  of  this  statement— Pipe  of  uniform  diameter  equivalent 
to  compound  system— General  formula— Numerical  example— Use  of 
logarithms  (foot  note)— Numerical  example  of  branch  pipe— Simplified 
method  -Numerical  examples— Relative  discharges  through  branches 
variously  placed — Discharges  determined  by  plotting — Caution  regard- 
ing results  obtained  by  calculation— Numerical  examples. 

CHAPTER  III. 

Numerical  example  of  a  system  of  pipes  for  the  supply  of  a  town — Es- 
tablishment of  additional  formulae  for  facilitating  such  calculations- 
Determination  of  diameters— Pumping  and  reservoirs— Caution  regard- 
ing calculated  results— Useful  approximate  formula— Table  of  5th 
powers— Preponderating  influence  of  diameter  over  grade  illustrated  by 
example. 

CHAPTER  IV. 

Use  of  formula  14  illustrated  by  numerical  example  of  compound  sys- 
tem combined  with  branches — Comparison  of  results — Rough  and 
smooth  pipes— Pipes  communicating  with  three  reservoirs— N  umerical 
examples  under  varying  conditions — Loss  of  head  from  other  causes 
than  friction— Velocity,  entrance  and  exit  heads— Numerical  examples 
and  general  formula  — Downward  discharge  through  a  vertical  pipe — 
Other  minor  losses  of  head— Abrupt  changes  of  diameter— Partially 
opened  valve — Branches  and  bends — Centrifugal  force — Small  import- 
ance of  all  losses  of  head  except  frictional  in  the  case  of  long 
pipes — All  such  covered  by  "  even  inches  "  in  the  diameter. 


E.  Sherman  Gould, 

M.  Am.  Soc.  C.  E. 

Consulting  and  Constructing  Engineer 
for  Water-Works. 

SCRANTON,    PA. 


HYDRAULIC    FORMULyE. 


CHAPTER  I. 


Flow  through  a  Short  Horizontal  Pipe'—  Effect  on  Velocity  of  Increased  Length— 
Frictional  Head—  Hydraulic  Grade  Lme—  Hydrostatic  and  Hydraulic  Preswire* 
—  Piezometric  Tubes  —  Result  of  liaising  a  Pipe  Line  Above  the  Hydraulic  Grade 
Line  —  Why  the  Water  Ceases  to  Rise  in  the  Upper  Stories  of  the  Houses  of  a 
Town  when  the  Consumption  is  Increased—  Influence  of  Inside  Surface  of  Fipes 
upon  Velocity  of  Flow—Darcy's  Coefficients—  Fundamental  Equations—  Length 
of  a  Pipe  Line  usually  Determined  by  its  Horizontal  Projection—  Numerical  Ex- 
amples <>f  Simple  and  Compound  Systems. 

Let  us  suppose  a  reservoir  of  large  relative  area  and  capacity 
to  be  tapped  near  its  bottom  by  a  horizontal  cylindrical  pipe,  of 
which  the  length  is  equal  to  about  three  times  its  diameter. 

If  there  were  no  physical  resistance  to  the  flow,  the  velocity  of 
the  water  issuing  from  the  pipe  would  be  given  by  the  formula  for 
the  velocity  of  falling  bodies  : 


in  which  V—  velocity  in  feet  per  second,  g  =  the  acceleration  due 
to  gravity  =  32.2  ft.,  and  H  =  the  height,  expressed  in  feet,  of  the 
surface  of  the  water  in  the  reservoir  above  the  center  of  the  pipe. 

Observation  shows,  however,  that  in  the  case  cited  the  velocity 
of  discharge  is  equal  only  to  that  theoretically  due  to  a  height  of 
about  two-thirds  of  H,  that  is  : 


8  PRACTICAL    HYDRAULIC   FORMULAE. 

The  remaining  third  of  the  height  is  consumed  in  overcoming 
the  resistance  offered  to  entry  by  the  edges  of  the  orifice  to  the  in- 
flowing vein  of  water.  The  head  necessary  to  overcome  the  resis- 
tance to  entry  is,  therefore,  about  one-half  of  that  necessary  to 
produce  the  velocity  of  flow. 

If  the  length  of  the  pipe  should  be  increased  progressively  and 
indefinitely,  the  velocity  would  be  found  to  diminish  inversely  as 
the  square  root  of  the  length.  It  would  correspond,  therefore,  lo 
a  smaller  and  smaller  percentage  of  the  total  head  H.  The  resis- 
tance to  entry  diminishes  directly  as  the  velocity,  and  the  head 
necessary  to  overcome  it  is  always  equal  to  about  one-half  of  that 
necessary  to  produce  the  given  velocity  as  calculated  by  the  laws 
of  falling  bodies. 

As  the  length  of  the  pipe  (always  supposed  to  remain  hori- 
zontal) increases,  and  the  velocity  of  discharge  diminishes,  the 
sum  of  these  two  heads,  i.  e.,  one  and  a  half  times  that  necessary 
to  produce  the  actual  velocity,  is  no  longer  equal  to  the  total  head 
H,  as  we  have  seen  to  be  the  case  when  the  length  of  the  pipe  is 
only  about  three  diameters.  What  then  becomes  of  the  remainder 
of  H  ?  It  is  consumed  in  overcoming  the  frictional  resistances  en- 
gendered by  contact  of  the  moving  water  with  the  inside  surface  of 
the  pipe.  When  the  pipe  is  very  long,  and  the  velocity  therefore 
relatively  low,  the  sum  of  the  velocity  and  entrance  heads  is  small 
and  by  far  the  greater  part  of  the  total  head  is  required  to  force 
the  water  through  the  pipe  against  the  opposition  offered  to  its 
flow.  In  such  cases,  which  are  those  occurring  most  generally  in 
practice  when  water  is  conveyed  from  a  reservoir  for  the  supply  of 
a  town,  the  velocity  and  entrance  heads  are  commonly  ignored,  and 
the  total  head  H  is  supposed  to  be  available  for  overcoming  the 
frictional  resistances.  As  this  occasions,  however,  an  error— al- 
though generally  a  very  small  one— in  the  wrong  direction,  judg- 
ment is  required  in  exercising  this  latitude,  Later  on  we  will  re- 
vert to  this  point,  but  for  the  present,  we  will  consider  only  fric- 


PRACTICAL     HYDRAULIC    FORMULAE. 


9 


tional  resistances,  particularly  since — and  indeed  because — in 
practice  our  assumed  data  are  almost  always  sufficient  to  afford 
an  ample  margin  to  cover  the  neglected  factors. 

In  what  precedes  we  have  considered  a  horizontal  pipe  issuing 
from  a  reservoir  in  which  the  surface  of  the  water  is  maintained  at 
a  constant  level.  In  practice  these  conditions  rarely  obtain. 


Fig.  i. 


Suppose  a  system,  such  as  is  shown  by  Fig.  1,  consisting  of  a 
reservoir  and  pipe  line  of  varying  and  contrary  slopes.  As  the 
level  of  the  water  in  the  reservoir  would  be  subject  to  fluctuations, 
and  liable  at  times  to  be  greatly  drawn  down,  it  is  customary  to 
consider  the  surface  of  the  water  as  standing  at  its  lowest  possible 
level,i.  e.,  the  mouth  of  the  pipe.  In  Ibis  case,  the  value  of  H 
would  be  equal  to  the  difference  of  level  of  the  two  extremities 
a  and  b  of  the  pipe,  and  the  line  a  b  joining  the  centers  of  the  two 
ends  would  form  what  i  called  the  hydraulic  grade  line,  the  estab- 
lishing of  which  is  the  first  step  to  be  taken  in  laying  out  a  system 
or  water  supply. 


10  PRACTICAL    HYDRAULIC    FORMULAK. 

Suppose  that  at  the  points  c,  d,  and  e  vertical  tubes,  open  at 
their  upper  ends,  were  connected  with  the  pipe.  The  water,  when 
flowing  freely  from  the  end  b  of  the  pipe  would  rise  in  each  of 
these  tubes  to  about  the  height  of  the  hydraulic  grade  line  at  these 
points,  and  if  branches  were  connected  at  the  points  c,  d, 
and  e,  they  would,  when  closed,  sustain  a  pressure  upon  their 
gates  equal  to  the  head  comprised  between  the  gates  and  the 
grade  line,  If  the  gates  were  open,  the  branches  would  discharge 
water  under  heads  equal  to  the  difference  of  level  of  the  hydraulic 
grade  line  at  the  point  of  embranchment  and  their  remote  ex- 
tremities, less  a  certain  amount  depending  upon  the  volume  dis- 
charged, which  will  be  spoken  of  hereafter. 

At  rf,  where  the  top  of  the  pipe  just  touches  the  grade  line, 
there  would  be  no  pressure  at  all  when  the  water  was  flowing 
through  the  pipe,  except  the  very  small  amount  due  to  the  depth 
of  water  in  the  pipe  itself. 

If  the  end  b  should  be  closed  so  that  there  was  no  movement 
of  water  in  the  pipe,  the  water  would  rise  in  the  tubes,  if  they 
were  long  enough,  until  it  stood  at  the  same  level  as  the  water  in 
the  reservoir  and  the  pressures,  at  c,  d,  and  e,  would  be  equal  to 
the  head  comprised  between  these  points  and  the  level  of  the 
water  in  the  reservoir.  This  latter  is  called  the  hydrostatic  press- 
ure, or  simply  the  static  pressure,  and  the  former  the  hydraulic 
pressure,  at  these  points. 

The  tubes  spoken  of  are  known  by  the  name  of  piezometric 
tubes. 

The  importance  of  correctly  establishing  the  hydraulic  grade 
line  is  illustrated  by  reference  to  a  case  such  as  is  shown  in  Fig.  2, 
in  which  ihe  pipe,  at  the  point  c.  rises  above  the  grade  line  a  b, 
To  explain :  It  will  be  readily  deduced  from  what  has  been  al- 
ready said  in  reference  to  horizontal  pipes  that  the  velocity  of  flow, 
and  consequently  the  delivery,  of  a  pipe  increases  with  the  steep- 
ness of  its  slope.  In  this  case  the  pipe  a  b  is  divided  into  two  parts- 


PRACTICAL    HYDRAULIC    FORMULAE. 


11 


the  one  a  c  with  a  hydraulic  grade  line  flatter  than  a  b  and  the 
other  c  b  with  one  steeper  than  a  b.  The  delivery  of  the  entire 
system,  if  the  pipe  were  of  the  same  diameter  throughout,  would 
be  governed  by  the  flatter  portion  a  c,  and  the  portion  c  b  would 
be  capable,  in  virtue  of  its  steeper  slope,  of  discharging  a  greater 
volume  of  water  than  it  could  receive  from  a  c.  Consequently  it 
would  act  merely  as  a  trough  and  would  never  run  full,  and 
if  a  piezometric  tube  were  placed  in  it  at  d,  for  instance  no  water 
would  rise  in  the  tube,  and  no  pressure  be  exerted. 


Fig.  2. 


It  is  very  important,  therefore,  in  locating  a  pipe  line  that 
the  pipe  should  nowhere  rise  above  the  hydraulic  grade  line.  The 
full  amount  of  water  could  indeed  be  carried  over  the  high  point 
c  by  means  of  syphonage,  but  this  expedient  is  not  resorted  to  in 
practice.  Should  the  nature  of  the  ground  require  such  a  location 
as  that  shown  in  Fig.  2,  it  would  be  necessary  to  increase  the  dia- 
meter of  the  pipe  between  a  and  c,  so  that  it  would  deliver  the  re- 


12  PRACTICAL    HYDRAULIC    FORMULAE. 

quired  volume  under  the  reduced  head,  and  to  diminish  that  be- 
tween c  and  b,  so  that  it  should  only  deliver  the  same  volume 
under  its  increased  head,  and  therefore  run  full.  The  calculations 
necessary  to  determine  the  proper  diameters  will  be  shortly  de- 
veloped. 

Should  the  axis  of  the  pipe  coincide  exactly  with  the  hydraulic 
grade  line  ab,  the  pipe  would  run  full  (provided  the  feed  were  suffi- 
cient) but  would  be  under  no  pressure,  and  no  water  would  rise  in 
piezometric  tubes  placed  on  any  part  of  the  pipe.  Moreover,  as 
the  slope  would  be  the  same  for  any  portion  of  the  pipe,  the 
velocity  and  delivery  would  be  unchanged,  whether  we  cut  the 
pipe  off  at  a  comparatively  short  length,  or  extended  it  indefi- 
nitely. 

As  a  further  and  very  interesting  practical  illustration  of  the 
effects  of  a  hydraulic  grade  line  of  varying  steepness,  let  us  con- 


Fig.  3. 


sider  (Fig.  3)  the  case  of  a  house  supplied  with   water   by  a  pipe 
communicating  with  a  reservoir. 


PRACTICAL    HYDRAULIC     FORMULAE.  13 

Suppose  the  pipe  to  be  just  sufficiently  large  to  furnish  a  cer- 
tain volume  of  water  per  hour  to  the  upper  story  of  the  house. 
If  now  a  larger  volume  were  required,  it  is  clear  that,  unless  we 
increase  the  diameter  of  the  pipe,  it  would  be  necessary  to  in- 
crease the  steepness  of  pitch  of  the  grade  line,  in  other  words,  to 
increase  the  head,  or  difference  of  level  between  the  reservoir  and 
the  point  of  discharge.  The  increased  volume  could  therefore  be 
only  drawn  from  a  lower  story. 

Or,  to  put  in  same  conditions  under  a  different  form,  suppose, 
as  before,  the  pipe  to  be  just  large  enough  to  supply  ttie  top  story 
of  the  house,  the  taps  on  the  lower  floors  being  closed.  Should 
they  be  opened,  it  is  evident  that  a  greater  amount  of  water  would 
be  discharged  from  them  than  from  the  upper  one,  because  they 
would  discharge  under  a  greater  head.  The  result  would  be  a 
diminished  flow,  or  perhaps  no  flow  at  all  on  the  top  floor,  and  an 
increased  discharge  of  water  at  a  lower  level. 

This  case  shows  why  the  water  ceases  to  rise  in  the  upper 
stories  of  the  houses  of  a  town  when  the  consumption  increases. 

It  has  been  found  by  observation  that  the  velocity  of  water 
flowing  through  pipes  is  greatly  affected  by  the  nature  of  their 
inside  surface,  increasing  with  the  smoothness  and  diminishing 
with  the  roughness  of  the  same.  By  direct  experiment,  coeffici- 
ents have  been  established  for  different  conditions  of  surface. 
It  na,s  also  been  found  that  these  coefficients  vary  slightly  with  the 
diameter  of  the  pipe,  a  pipe  of  a  certain  size  giving  a  greater  veloc- 
ity than  one  of  the  same  character  of  inside  surface  but  of  smaller 
diameter,  the  differences  becoming  smaller  as  the  diameters  in- 
crease. 

The  value  of  this  coefficient,  which  will  be  designated  through- 
out this  paper  by  C,  is  given  below  for  a  number  of  different  dia- 
meters and  for  two  classes  of  pipes, — those  which  are  clean  and 
smooth  on  the  inside,  and  those  which  are  rough  and  incrusted, 
the  difference  being  as  2  to  1.  As  all  pipes,  after  a  few  years  of 
service,  are  liable  to  become  more  or  less  roughened  and  ob- 


14 


PRACTICAL    HYDRAULIC   FORMULAE. 


structed  by  deposits  it  is  always  safer  when  calculating  the  proper 
diameters  of  a  permanent  water  supply,  to  assume  rough  pipes 
at  once,  although  diameters  thus  calculated  will,  for  perhaps  a 
number  of  years,  deliver  quantities  greatly  in  excess  of  the  de- 
sired amounts. 

The  coefficients  given  below  are  those  determined  experiment- 
ally by  DARCY.  Of  course,  in  the  subsequent  calculations  which 
will  be  made,  any  other  values  might  be  substituted  for  the  ones 
given.  It  is  well  to  remark,  however,  in  regard  to  the  coefficient, 
that  although  this  factor  is  a  controlling  one  in  the  calculation  of 
the  discharge  of  pipes,  it  is  useless  to  attempt  an  excessive  refine- 
ment in  establishing  its  value,  because  not  only  is  it  difficult  to 
determine  this  value  with  exactness  for  a  given  diameter  and  con- 
dition of  pipe,  but  this  condition,  and  even  the  diameter  of  the 
pipe,  is  liable  to  undergo  considerable  variation  in  the  same  pipe 
in  the  course  of  a  few  years. 


Diameter  in 
inches. 

3 

4 

6 

8 

10 
12 
14 
16 
24 
30 
36 
48 


TABLE    OF  COEFFICIENTS. 

Value  of  C  for 
rough  pipes. 

0.00080 
0  OOOT6 
0.00072 
0.00068 
0.00066 
0.00066 
0.00065 
0.00064 
0.00064 
000063 
0.00062 
0.00062 


Value  of  C  for 
smooth  pipes. 

0.00040 

0.00038 

0.00036 

0.00034 

0.00033 

0.00033 

0.000325 

0.00032 

0.00032 

0.000315 

0.00031 

0.00031 


In  all  the  following  calculations,  the  coefficient  for  rough  pipes 
will  be  used. 


PRACTICAL    HYDRAULIC    FORMULAE.  15 

The  two  fundamental  equations  relating  to  the  flow  of  water 
through  long  pipes  are  : 

D  x  H 

=  C  Fa  (1) 

L 

Q     =  A  V  (2; 

Equation  No.  2  will  generally  be  written  : 

/D  x  H 

Q  =  A  I/-  (3) 

V    C   x  L 

by  taking  the  value  of  V  from  (1). 

The  first  of  these  has  been  established  by  DARCY  ;  the  second 
is  based  upon  a  self-evident  proposition. 

In  these  equations : 

D  =  diameter  of  pipe  in  feet 
H  =  total  head  "    " 

L  =  length  of  pipe       "    " 
C  =  coefficient 

V  =  mean  velocity  in  feet  per  second 
Q  =  disc  harge  in  cubic  feet  per  second 
A  =  area  of  pipe  in  square  feet  =  D2  x  0.785 

The  above  two  formulae  solve,  directly  or  indirectly,  all  prob- 
lems relating  to  the  flow  through  long  pipes,  and  all  such  prob- 
lems must  be  brought  into  a  form  admitting  of  their  application, 
in  order  to  obtain  a  solution. 

H 

It  will  be  observed  that  —  is  the  rise  or  fall  per  foot  of  length 
L 

of  pipe,  and  is  therefore  the  natural  sine  of  the  inclination  of  the 
slope  to  the  horizon.     This  relation  is  frequently  used  under  the 

H 

form     /   =  —  .       Using  this  notation,  (1)  would  be  written  : 
L 

D  I  =  C  V? 


16  PRACTICAL    HYDRAULIC    FORMULAE. 

In  long  pipes  the  length  is  generally  taken  as  being  equal  to 
the  horizontal  distance  separating  the  two  ends  of  the  pipe,  as  the 
difference  between  this  distance  and  the  actual  length  of  the  pipe 
is  relatively  insignificant.  If,  however,  a  case  should  present  it- 
self in  which  this  difference  was  considerable,  the  actual  length 
of  pipe  should  be  taken.  Further  on,  an  extreme  case  of  this  kind 
will  be  given,  presenting  some  interesting  features. 

Some  practical  examples  of  the  use  of  these  formulae  will  now 
be  given.  In  all  that  follows,  the  resistances  of  entry,  exit,  and 
velocity  will  be  neglected,  and  the  total  head  will  be  considered  as 
available  for  overcoming  friction.  The  examination  of  cases  where 
the  above  factors  are  included  is  reserved  for  a  later  portion  of 
this  paper,  as  they  are  of  secondary  importance  when  dealing  with 
long  pipes. 

Example  1.— A  pipe,  1  ft.  in  diameter  and  1,000  ft.  long,  has  a 
total  fall  of  10  ft.  What  are  the  velocity  and  volume  of  its  dis- 
charge? 

Substituting  the  given  values  in  (1)  we  have  : 
1  x  10 


=  0.00066  V2 


1,000 

V  —  3.89  ft.  per  second. 

Using  this  value  of  Fin  (2),  we  have ; 
£  =  0.785  X  3.89 
Q  =  3.055  cu.  ft.  per  second. 

Example  2. —  Two  reservoirs,  having  a  difference  of  level  of 
water  surface  of  30  ft.  are  joined  by  a  pipe  3,000  ft.  long.  What 
should  be  the  diameter  of  the  pipe  to  deliver  16  cu.  ft.  of  water  per 
second  from  the  upper  to  the  lower  reservoir? 

Eliminating  V  between  (1)  and  (2)  we  have  : 

D  x  H  _    Q2 
L  X  G~  ~A* 


PRACTICAL  HYDRAULIC    FORMULAE.  17 

Observing  that  A  =  D*  0.785; 

D  x  H  <?2 


L  X  C         D*  X  0.616 
Whence 

£)»  x  Z  x  (7 


#  X  0.616 

If  we  knew  the  proper  value  of  the  coefficient  C  in  the  above 
equation,  it  could  be  immediately  solved,  and  the  value  of  D  ob- 
tained. But  C  varies  with  the  diameter,  and  the  diameter  is  as  yet 
unknown.  We  must,  therefore,  have  recourse  to  "  Trial  and 
Error  "  for  a  solution. 

Suppose  it  should  appear  to  us,  at  first  sight,  that  a  12-in.  pipe 
was  likely  to  be  of  the  proper  size.  We  therefore  take  C  =  0.00066, 
and  write  : 

256  X  3,000  X  0.00066 
D*  =  -  - 

30  X  616 

D6   =   27-70 
D  =  1.94  ft. 

From  this  we  see  that  the  pipe  should  be  nearly  2  ft.  in  dia- 
meter, and  as  we  have  taken  too  large  a  coefficient  (that  for  24  ins. 
-=0.00064),  we  are  sure  that  1.94  is  too  large.  As  pipes  are  never 
made  of  fractional  diameters,  the  above  value  of  D  would  be  taken 
•-=  24  ins.,  and  therefore  we  would  push  the  calculation  no  further. 
If  the  case  had  happened  to  be  one  requiring  minute  accuracy,  we 
would  recalculate  the  above  equation,  using  0.00064  for  the  value 
of  C.  The  result  would  be,  D  -  1.93  ft.  nearly,  differing  very 
slightly  from  the  value  already  obtained. 

The  above  examples  (which  are  those  commonly  occurring  in 
practice)  are  very  simple,  and  involve  only  the  direct  application 
of  the  fundamental  formulae.  Let  us  now  consider  cases  of  a  more 
complicated  character,  where  they  can  only  be  used  indirectly,  and 
where  a  certain  amount  of  judgment  and  tact  is  required  in  the 
preparation  of  the  data, 

Example  3.—  Suppose  a  reservoir  R  (Fig.  4)  containing  a  depth 


18  PRACTICAL  HYDRAULIC    FORMULAE. 

of  water  of  50  ft.  above  the  center  of  the  horizontal  pipe  A,  1  ft.  in 
diameter  and  1,000  ft.  long,  connected  by  a  reducer  with  another 
horizontal  pipe  B,  2  ft.  in  diameter  and  3,000  ft.  long.  It  is  required 
to  calculate  the  piezometric  head  h  at  the  junction,  from  which  the 
discharge  can  be  calculated,  and  the  hydraulic  grade  line  abc  es- 
tablished. 


Fig.  4. 

It  is  evident  that  the  24-in.  pipe  must,  under  the  head  h,  dis- 
charge the  same  quantity  per  second  as  the  12-in.  pipe,  under  the 
head  50 — h,  We  have  then  from  (3),  the  equality : 


=  0.785  I/1 


1000  x  0.00066 


Dividing  by  0.785,  squaring,  and  simplifying: 

h          50— h 
O.OV!  =      U.22 

whence 

h  =  4.17 

We  can  now  very  readily  get  the  discharge,  by  substituting  the 
value  4.17  for  h  in  either  member  of  the  above  equality.     Thus : 

Q  =  3.141/  -i^— : —  =  6-54  cu.  ft.  per  sec. 

Verifying  in  the  other  member — a  precaution    which  should 
never  be  neglected — we  obtain  the  same  result. 

It  is  evident  that  the  diameter  of  B  may  be  assumed  so  large 


PRACTICAL    HYDRAULIC    FORMULAE. 


19 


that  no  value  of  h  can  be  found  to  satisfy  the  condition  that  both 
pipes  shall  run  full  with  the  given  height  of  water  in  the  reservoir. 
In  such  a  case  the  pipe  B  serves  only  as  a  trough  to  receive  the 
water  discharged  through  A  under  a  head  of  50  ft. 

Suppose  that  in  the  above  example,  the  places  of  the  two  pipes, 
A  and  B  should  b3  changed.  Evidently  we  should  have  : 

h  =  45.83 

This  piezometric  height  would  give,  with  the  transposed  posi- 
tion of  the  pipes,  the  same  discharge  as  before,  the  only  difference 
being  a  notable  change  in  the  hydraulic  grade  line.  If  the  pipes 
were  tapped  by  branches,  the  greater  elevation  of  the  grade  line  in 
this  case  would  bring  a  much  greater  pressure  upon  the  branches, 
enabling  them  to  deliver  water  at  a  higher  level  than  in  the  first 
position  of  the  pipes. 

~r~        ----.  -r^  ^ 


16* 


500  800  liOO  o'JO 

Fig  5. 

The  above  example  may  be  extended  so  as  to  cover  cases  where 
pipes  of  several  different  diameters  are  used.  Thus,  suppose  a 
system  of  pipes,  such  as  is  shown  in  Fig.  5,  where  a  reservoir  with 
a  head  of  50  ft.  of  water,  as  before,  is  taoped  by  a  horizontal  line  of 
pipes,  consisting  in  order  of  500  ft.  of  12-.in.,  800  ft,  of  16-in.,  1,400  ft. 
of  8-in.  and  600  ft.  of  6-in.,  pipe. 

This  example  may  be  worLed  in  the  same  way  as  the  previous 
one,  by  getting  equations  for  h,  h',  and  h"  expressed,  by  substitu- 
tion, in  terms  of  h.  But  it  wiil  be  easier  to  treat  the  question  in 
another  way,  which  will  also  exhibit  the  further  resources  which 
we  have  at  our  disposal  in  solving  hydraulic  problems. 


20 


PRACTICAL   HYDRAULIC    FORMULAE. 


Since  each  section  of  pipe  must  discharge  equal  volumes  in 
equal  times,  it  is  evident  that  the  respective  velocities  of  flow  must 
vary  inversely  as  the  areas  of  the  pipes.  These  areas  vary  as  the 
squares  of  the  different  diameters.  Designating,  therefore,  by  V 
the  lowest  rate  of  velocity,  i.  e.,  that  of  the  water  passing  through 
the  largest  pipe  (the  16-in.  one),  we  obtain  the  relative  velocities  in 
the  other  pipes  by  multiplying  Fby  the  ratio  of  the  square  of  the 
diameter  of  the  largest  pipe,  to  the  squares  of  the  other  diameters. 
It  will  be  convenient  to  form  the  following  table  ; 


Lengths  in  ft. 

Diameters  in  fr. 

Velocities  in  ft. 
per  second. 

500 
800 
HOO 
600 

1 
1H 

% 

y* 

1.78  V 
V 
4  V 
7.11  V 

Beginning  at  the  lower  end  of  the  system,  that  is  with  the  6-in. 
pipe,  and  employing  formula  (1)  in  which  h  and  Fare  the  unknown 
quantities,  we  have. 


1  h 

—  X  =  0.00072  X  (7.11)2  X   F2 

2  600 


whence: 

again : 

•whence: 

similarly: 

whence: 

Finally: 


2  (h'  —  h) 

V 

3  1400 


4 
3 


h  =  43.68  F2 

.68   V 

\  =  0.00069  X  (4)2  X  F2 
0 

h'  ~  63  86  F2 

36   F2\ 

=  0.00065  X   F2 


2  /h'  —  43.68    F2  \ 

=  -  x  I  -  -)  =  o. 

3  \  1400  / 

}.. 


50  —  h" 


k"  —  66  86   F 

800 
h"  =  67.25   F2 


50  —  67.25  F2 

—  =  0.00066  X0.78)2  F2 
500 


whence:  F2  =  0.7321 

F      0.8556  ft.  per  second 
Substituting  this  value  of  V*  in  the  above  equations  : 

h  =31.9H  fr. 
h'  =  48.95  " 
h"  =  49.23  " 


PRACTICAL    HYDRAULIC    FORMULAE.  21 

We  also  get  the  velocities  in  the  different  pipes,  thus : 


6  incn,  velocity  =  7.11  X  O.P56  =  6.086 

8     "  "  =      4  X  0.856  =  3.424 

16    "  "  =       1  X  0.856  =  0.856 

12     "  "  =  1.78  X  0.856  =  1.524 


The  work  can  be  checked  by  using  the  above  values  of  h,  h' 
and  h",  along  with  the  other  data,  in  (1),  and  obtaining  the  veloci- 
ties in  this  way. 

Thus,  beginning  with  the  6  inch  pipe ; 


=  0,00072  F2 


V  = 

16.97 


=  0.00069  F1 

3    1400 

F  =  3.42 
4    0.28 

—  X  =  0.00065  F"2 

3     800 

V"  =  0.85 

0.77 

1  X  =  0.00065  F"2 

500 

V"  =  1-53 

A  very  close  agreement  throughout. 

In  the  above  calculations  the  decimals  have  been  carried  out 
further  thau  would  ordinarily  be  necessary  in  practice.  It  was 
done  in  the  present  instance  in  order  to  avoid  discrepancies  in 
checking. 

W7e  have  another  check,  in  the  volumes  discharged.  Thus 
the  discharge  through  the  6-in.  pipe,  with  the  given  velocity  is  by 
(2). 

Q  =  0  195  X  6.086 
Q  =  1.19  cubic  ft.  per  second. 

All  the  other  pipes  should  have  an  equal  discharge,  for  instance 
the  12-in.  pipe  gives: 

Q  =  0.78  X  1.523 

Q  =  1.19  cubic  ft.  per  second. 


CHAPTER    II. 

Calculations  are  the  Same  for  Pipes  laid  Horizontally  or  on  a  Slope— Qualification 
of  this  Statement — Pipe  of  Uniform  Diameter  Equivalent  to  Compound  System 
—  General  Formula— Numerical  Example— Use  of  Logarithms  (foot  note)— Nu- 
merical example  of  branch  pipe— Simplified  method — Numerical  t-xamples — 
Relative  discharges  through  branches  variously  placed — Discharges  determiiied 
by  plotting— Caution  regarding  results  obtained  by  calculation — Numerical 
examples. 

In  the  preceding  examples  a  series  of  horizontal  pipes  has 
been  considered,  the  head  being  produced  by  an  elevated  reservoir 
placed  at  one  end.  The  results  would  have  been  identical,  how- 
ever, if  the  head  had  been  produced  by  the  pipes  being  laid  upon 
a  slope,  provided  the  difference  of  level  between  the  two  extremi- 
ties remained  the  same,  for  the  velocities  and  hydraulic  grade  line 
would  remain  unaltered.  The  pressure  in  the  pipes  would  vary 
however,  according  to  their  distance  below  the  hydraulic  grade 
line,  the  pressure  being  measured  at  any  given  point  in  the  pipe 
line,  by  the  vertical  distance  between  such  point  and  the  grade 
line.  If  the  pipes  were  laid  exactly  upon  the  hydraulic  grade  line 
there  would  be  no  pressure  at  all  in  the  pipes,  and  if  they  rose  at 
any  point  above  it,  there  would  be  either  no  flow  or  a  diminished 
one,  unless  syphonage  were  resorted  to. 

In  order  to  make  this  point  very  plain,  we  will  consider  the 
same  system  of  pipes  as  that  used  in  the  last  example,  but  laid  as 
shown  in  Fig.  6,  the  upper  extremity  being  fed  by  a  constant 
supply,  with  only  head  enough  to  overcome  resistance  to  entry, 
and  produce  initial  Telocity,  which  will  be  treated  of  further  on. 


PRACTICAL    HYDRAULIC   FORMULAE. 


23 


Calculating  precisely  as  before,  we  get  the  same  hydraulic  grade 
line,  unbroken  by  the  rising  grade  of  the  last  200  ft.  of  6-in.  pipe. 


Fig,     6. 


It  is  sometimes  desirable  to  ascertain  the  uniform  diameter  of 
a  pipe  which  shall  be  equivalent  to  a  series  of  pipes  of  different 
diameters,  such  as  we  have  just  been  studying.  This  may  be  done 
by  an  application  of  formula  (4),  which,  for  this  purpose  is  written 
in  the  following  form  ; 


G  Q* 

0.616 


L 

x  — 
D* 


As  an  example,  let  us  calculate  the  diameter  of  a  single  pipe, 
of  the  same  total  length  and  fall  as  the  series  of  pipes  which  we 
have  just  had  under  consideration,  and  capable  of  discharging  an 
equal  volume.  We  will  first  establish  the  general  formula  for  all 
such  problems,  expressing  the  difference  obpiezometric  level  be- 
tween the  two  ends  of  each  pipe  respectively,  by  7iT  h?  h?  h±,  etc., 
their  respective  lengths  by  I  I?  l^  I?  etc.,  their  respective  diameters 
by  cZT  c/2  t/3  (??  etc..  and  their  respective  coefficients  by  CT  c?  c?  c¥  etc. 
commencing  with  the  lower  end.  We  will  express  the  total  length 
by  L,  the  total  difference  of  level  by  H,  the  unknown  diameter  by 
D,  and  its  coefficient  by  C. 


24  PRACTICAL    HYDRAULIC  FORMULAE. 

Now,  observing  that   the  quantity  discharged  per  second  by 
each  pipe  is  the  same,  we  have  the  4  equations. 

c,  Q2       /' 

7l!    =     -     X     - 

0.6.6         d? 

c2  Q2        J,1 

/*„  =    -   X    - 

0.616          dO 

c3  Q2        ?3 

//3  =  -  x  - 

0.616  da 


0616         df 

Adding,  and  observing  that  the  sum  of  the  partial   heads  hi  7t2 
7i7  7i?  equals  -H,  we  have  : 

-    Q2   I  €l  ll    '    C2/2       £i_fi    ,   S 
0.616  \  dr  d25  d35  d4 

but  we  have  also  the  equation 

(7Q2    '   L 
H  =—  —  x  - 

0.616        J>5 

whence,  suppressing  the  common  factor; 

C  L  d  h  C2  Za  C3  ^3  04  ?v 

-  =    --  1  ---  1  ---  1  ---  (5) 

D"          dis  d25  d3s         d  4  B 

The  above  is  the  general  formula. 
Substituting  the  special  values  of  our  example  : 

3300  0.33        0.52        0.96fi        0.432 

-  X  (7=  --  1  ---  1  ---  1-  - 
IP  1K      (4  3)5      (2  3,s        (1-2JB 

Giving  a   preliminary  approximate  value  to  C  of  0.00066,  we 
have 

2-178 


=  0-33  +  0.123  +  7.335    H    13.824 


=  0.1007 
=  0.63 


This  value  of  D  indicates  a  practical  diameter  of  Sins. 

in  order  to  check  this  value,  we  mav  write  (4)  under  the  form  : 


Q  =  |/  J>X     If  V    0.616 
7,  X  C 


PRACTICAL    HYDRAULIC    FORMULAE.  25 

Substituting  given  values : 


0.1007  X  50  X  0.61C 
V    ~   y    "  2.178 

^  =  1.193  cu .  ft.  per  second. 

thus  proving  the  correctness  of  the  work. 

These  calculations  can  be  abridged,  and,  in  many  cases,  suffi- 
cient accuracy  secured  by  adopting  a  mean  common  value  for  C. 
If  we  do  so  in  the  present  case,  C  becomes  a  common  factor,  and 
disappears  from  the  calculation,  (5)  becoming 

L        it         h        h 

= 1-    —    4-     —  etc.  (5)  bis 

D*       d,"       dj         <tf 

If  this  equation  be  worked  out  for  the  above  given  values,  we 
have : 

D—  0.64 

or  8  ins.  as  before. 

It  will  be  observed  that  this  process  might  have  been  used 
with  advantage  in  the  previous  example,  by  ascertaining  the  dis- 
charge of  an  equivalent  pipe,  and  then  calculating  the  heads 
necessary  to  produce  this  discharge  through  the  different  pipes. 

In  calculating  fifth  powers  and  roots,  a  table  of  logarithms  is 
almost  indispensable.  If  none  is  at  hand  a  table  of  squares  and 
cubes  is  of  some  use,  remembering  that  a  number  can  be  raised  to 
the  fifth  power  by  multiplying  together  its  square  and  cube.  Fifth 
roots,  in  the  absence  of  logarithms,  can  only  be  extracted  by 
"trial  and  error,"  using  the  above  rule  for  fifth  powers.* 

Example  4th.  A  horizontal  pipe  (Fig.  7).  48  ins.  in  diameter 
and  2,000  ft.  long,  issues  from  a  reservoir  in  which  the  surface  of 
the  water  is  maintained  at  a  constant  height  of  50  ft.  above  the 
center  of  the  pipe.  Midway,  this  pipe  is  tapped  by  a  branch  pipe 
24  ins.  in  diameter  and  500  ft.  long,  with  a  rising  grade  of  4  ft  in  500. 


*  All  hydraulic  calculations  are  greatly  facilitated  by  tbe  use  of  logarithms, 
and  thoee  engaged  in  making  such  calculations,  should  not  fail  to  familiarize 
themselves  with  the  use  of  these  powerlul  auxiliaries  to  arithmetical  work. 


26 


PRACTICAL   HYDRAULIC    FORMULAE. 


What  is  the  piezometric  head  h  at  the  junction,  and  what  the  dis- 
charge from  each  pipe?* 

It  is  evident  that  the  48-in.  pipe  above  the  junction  must,  with 
the  head  50  —  h,  discharge  as  much  water  per  second  as  the  com- 


^•          500 


00 


Fig.  7. 


bined  discharge  of  the  48-in.  pipe  beiow  the  branch  with  the  head 
/*,  and  the  24-in.  pipe  with  the  head  /j—  4.  From  (3).  which  in  this 
case  will  perhaps  be  the  most  convenient  equation  for  quantity 
though  that  derived  from  (4)  is  frequently  useful,  we  have  : 


0=12.56 


4  (50  —  ft) 
1000  X    0  000(52 


«'    =  3.H|/  r 

which,  put  in  equation,  give: 

12.56   | 


/     2   (h—  4) 


<5 


1000    X  O.OOC62 


12.5fi4/I  *  l> 

V        1000    X  0.00*62 


+3  14 


500    X    0-00064 


*  With  these  given  lengths  and  diameters,  the  above  system  does  not  propetly 
come  under  the  classification  of  "lone1  pipes."  As  the  present  object  i«  only  to 
exemplify  methods  of  calculation,  the  example  is  equally  good. 


PRACTICAL    HYDRAULIC    FORMULAE.  27 

The  coefficients  0.00062  and  0.00064  are  so  nearly  equal  that  we 
may,  in  the  following  calculations,  discard  them  as  common 
factors.  Dividing  by  3.14  and  striking  out  also  the  common  fac- 
tors Tifor  and  55^,  wr  have  simply: 


4  >/50     —   h  =     4  Vh.         +       \/h      —      4 
Squaring:  £00  —  16  h  =  16  h  +  h  —  4  +  8  V  h9  —    4h 


which  gives  ;  33/<  =804  —  8  v/ft2  —  4/i 

Neglecting,  for  a  first  approximate  value  of  h  the   quantities 
affected  by  the  radical : 

33ft   =  804 

Neglecting  decimals : 

ft   =  24. 

Substituting  this  value  for  h  under  the  radical : 


33ft  =  804  —  8    V  576  —  96 

which  gives,  always  neglecting  decimals,  a'  second  approximate 
value : 

ft  =  19. 

A  third  and  fourth  approximation  give  respectively  li  =  20.3 
and  h  =  20. 

We  will  take  20.1  as  very  near  the  true  value. 

Substituting   20.1  in  place  of    h  in    the  equations  giving  the 
quantities  discharged,  we  have  : 


- 12  ™      4    '  29-9 

-  12.56  j/  _ 

*  0.6-2 


0.62 


=lt-,,5 


We  have  thus; 


28  PRACTICAL    HYDRAULIC    FORMULAE. 

The  above  method  gives  directly  the  true  value  of  h;  but  it 
involves  tedious  figuring,  even  in  our  example,  which  happens  to 
admit  of  many  simplifications  owing  to  the  number  of  common 
factors.  It  will  be  easier,  and  often  shorter,  to  obtain  the  value  of 
h  by  first  assuming  one  which  we  judge  likely  to  be  near  the  truth, 
calculating  what  discharge  it  would  give  from  the  two  branches, 
and  then  calculating  the  head  necessary  to  discharge  the  same 
quantity  from  the  single  pipe  above  the  branch.  Then,  comparing 
the  total  height  thus  obtained  with  the  known  height  of  the  water 
in  the  reservoir,  we  can  deduce  the  true  value  of  h  by  a  proportion. 

Let  us  apply  this  method  to  the  above  example.  We  know  at 
once  that  h  must  be  less  than  25,  because  that  would  be  its  value  if 
the  24-in.  branch  were  closed.  Supposing  we  judged  that  22  ft 
would  be  about  correct.  We  then  have  to  solve  the  two  equations  : 


/  4  X  2: 
=  12.56  /*/   - 

r         n  fi2 


22 

149.60 
0  62 


'2X18 
0.32 


3. 14 1/ =33.30 


also,  for  the  equal  discharge  through  the  48-in.  pipe  above  the 
branch,  squaring  (3),  we  have  : 

(182.90)2  X0.62 

h  = =  32.87 

(12-56)2X4 

This  height,  added  to  22,  the  assumed  value  of  h,  gives  a  total 
height  of  54.87  ft.  as  against  50  ft.,  the  actual  total  height.  By  pro- 
portion we  have : 

h         so 

22         54.87 

This  value  of  h  agrees  with  that  already  found. 

If  the24-in.  branch  were  closed,  we  should  have  for  the  dis- 
charge : 

/4X60 

Q  =  12.56  \/  —  =  159.51 

f         1.24 


PRACTICAL    HYDRAULIC    FORMULAE.  29 

When  the  24-in.  branch  was*  open,  we  had  a  total  discharge  of 
174.73  cu.  ft.  per  second.  There  is  an  increase,  therefore,  of  about 
9i  per  cent,  by  opening  the  branch. 

Let  us  now  see  what  the  discharge  would  be  if  the  branch  were 
placed  only  500  ft.  from  the  reservoir,  instead  of  1,000  ft.,  all  the 
other  conditions  remaining  the  same. 

•  We  will  assume  h  =  33  ft.  and  solve  the  two  equations 


/          4X33 

if =  149.; 

'     i  son  v  n  finnfta 


q  =  12.56  47 =  149.5 

1500  X  0.00062 


/  2  X  29 
0.32 


g'  =3.14  4       =  42.3 

"  0  30! 


(191.8)*  X  0.31 

alSO  h' = =  18.07 

(12. 56)*  X  4 

giving  a  total  height  of  51.07  as  against  50.    Reducing : 

h  50 

33        51.07 
h.  =--32. 3 

Using  this  value,  instead  of  the  assumed  one,  we  have  : 


4X177  4X  32.3  -j   -   28.3 

12.564 =  12.564 h  3.14  4 

0.31  0.93  0.32 

'189.83  =  148.03  +  41.76 

very  nearly. 

As  compared  with  the  discharge  when  the  24-in.  branch  is 
closed  this  shows  a  gain  of  19  per  cent.,  just  double  the  gain  when 
the  branch  was  located  at  the  center  of  the  pipe. 

Supposing  now  that  the  branch  were  placed  1,500  ft.  from  the 
reservoir.  Assuming  10  ft.  as  a  probable  value  of  h.  we  have : 


4  X  10 

32.564 =142.46 

POO  X  0.00062 


fj'  =3.14  4      =19.23 

0.32 


30  PRACTICAL    HYDRAULIC    FORMULAE. 

(161- 7)2  X  0.93 

also:  h' = =3853 

(12.56)2  X4 

h          5.i 
By  proportion 

10         4  8  53 

//  =  10.30 

Using  this  value  instead  of  the  assumed  one  : 


/  4X39.7  /  4X10.3  /  2  X  6. 

12.56.4       --  =12.56  4/    ---  h  3.14  A/    - 


0.93  0.32  0.32 

164.13  =144-57  +  19.68 


very  nearly. 


As  compared  with  the  discharge  when  the  24-in.  branch  is  closed 
this  shows  a  gain  of  not  quite  3  per  cent.,  which  is  in  marked  con- 
trast to  the  gain  when  the  branch  was  only  500  ft.  from  the  reser- 
voir, being  less  than  one-sixth  of  the  gain,  in  that  case. 

It  will  be  interesting  to  study  a  little  more  in  detail  the  question 
of  relative  discharges.  We  have  seen  that  when  there  is  no  branch 
open  on  the  48-in.  pipe,  its  discharge  is  159.51  cu.  ft.  per  second, 
The  24-in.  branches,  wherever  placed,  increase  the  total  discharge, 
but  diminish  that  in  the  48-in.  pipe,  below  the  branch.  By  com- 
paring the  above  quantities,  it  will  be  perceived  that  the  flow  from 
the  48-in.  pipe  is  diminished  approximately  by  that  proportion  of 
the  quantity  flowing  through  the  24-in.  branch  which  is  represented 
by  its  proportionate  distance  from  the  reservoir.  Thus,  when  the 
branch  is  1,500  ft.,  or  three-quarters  of  the  length  of  the  48-in.  pipe 
from  the  reservoir,  as  in  the  last  case,  its  discharge  is  19.62  cu.  ft. 
per  second.  Three-quarters  of  this  quantity  is  14.715,  which,  sub- 
tracced  from  159.51,  leaves  144.795,  or  very  nearly  that  of  the  48-in  - 
pipe  below  the  branch,  as  determined  by  calculation. 

In  the  same  way  half  of  the  discharge,  when  the  branch  is  situ- 
ated half  way  from  the  reservoir,  subtracted  from  159.51,  gives  also 
very  nearly  the  amount  discharged  below  the  branch.  When  the 


PRACTICAL    HYDRAULIC    FORMULAE. 


31 


branch  is  500  ft.,  or  one-quarter  of  the  total  distance  from  the  reser- 
voir, one  quarter  of  its  discharge  taken  from  159.51  gives  very 
closely  the  discharge  as  calculated  for  the  48-in.  pipe  below  the 
branch. 

Let  us  now  take  an  extreme  position  for  the  branch,  and  sup- 
pose it  placed  close  to  the  reservoir,  so  that  there  is  practically  no 
portion  of  the  48-in  pipe  between  it  and  the  reservoir.  There 
will,  therefore,  be  no  part  of  the  flow  from  the  branch  subtracted 
from  that  of  the  main  oipe,  and  the  two  will  each  discharge  the 
same  quantity  as  if  the  other  were  not  there.  That  is,  the  48-in 
pipe  will  discharge  159.51,  and  the  24-in.  53.24  cu.  ft.  per  second. 

If  we  should  take  another  extreme  position  for  the  branch,  and 
suppose  it  placed  at  the  end  of  the  48  in  pipe.it  is  obvious  that, 
with  its  assumed  rising  grade  of  4-ft.  in  500.  it  would  discharge  no 
water  at  all.  A  position  could  be  found  by  trial  where  it  would 
just  cease  to  discharge  water,  but  for  the  object  of  the  present  in- 
vestigation this  is  not  necessary. 


Pig.    8. 

If  the  above  results  are  plotted,  as  in  Fig.  8,  a  very  instructive 
diagram  is  obtained.    The  successive  500  ft.  lengths  being  laid  off 


32  PRACTICAL  HYDRAULIC    FORMULAK. 

as  abscissae,  and  the  discharges  measured  upon  the  corresponding 
ordinates,  it  will  be  seen  that  their  extremities  all  lie  nearly  in  the 
same  straight  line.  If,  therefore,  the  discharges  for  any  two  posi- 
tions of  the  branch  be  calculated,  and  a  straight  line  drawn  pass- 
ing through  their  extremities,  the  discharge  for  any  other  position 
of  the  branch  can  be  obtained  by  erecting  an  ordinate  at  the  given 
point  to  the  straight  line,  and  the  flow  through  the  main  also  ob- 
tained by  subtracting  the  proper  portion  of  that  of  the  branch. 

In  practice,  when  making  calculations  similar  to  those  under 
consideration,  one  error  must  be  carefully  guarded  against 
namely,  the  supposing  that  the  actual  results  will  be  exactly  as 
calculated.  The  chief  value  of  these  calculations  lies  in  the  fact 
that,  they  furnish  pretty  trustworthy  relative  results,  that  is,  they 
establish  fairly  well  in  practice  the  fact  that  if  a  certain  pipe  de- 
livers a  certain  volume  of  water  in  a  certain  position,  it  will  de- 
liver a  certain  greater  or  less  amount  in  another.  The  actual 
amounts,  in  either  case,  cannot  be  surely  determined,  as  they  de- 
pend upon  so  many  varying  circumstances  about  which,  even  when 
aware  of  their  existence,  we  have  no  exact  date. 

Let  us  next  suppose  a  system  in  which  the48-in.  pipe  is  tapped 
every  500  ft,  by  a  24-in.  pipe,  500  ft.  long,  laid  as  before  with  a  grade 
of  4  ft.  in  500.- 

Assuming  a  height  of  9  ft.  for  the  piezometric  column  h  nearest 
the  tree  end  of  the  pipe  we  have  : 


A  X  9    .  ,  /2  X  5 


12.664 /<        y  +  8.144/'  =  12,56  4/4  (h- 9) 

\     0.31  V     0.32  ^3T~ 

Since  the  denominators  under  the  radicals  are  so  nearly  equal 
we  may  cancel  them,  and  making  other  simplifications,  write  : 


V  9  +  7  r  10  = 

Whence:  /</  =  20.52 


\      H  52   +  -  |/  33.04    =:      J/   h"  -  20. 


20.52 


PRACTICAL    HYDRAULIC    FORMULAE.  38 

h"  =  37.43 


Also:  1       16.91      +  %  \     66.85    =  A/    h'"  —  37.43 


Bv  propoitioii  we  have: 


h'"  =  63.79 
h          50 


9        63.97 
It  =  7.05 

As  the  value  of  h'"  =  63.79  differs  considerably  from  the  true 
value  =  50,  and  as  the  above  proportion  is  not  exactly  absolute, 
particularly  in  a  somewhat  complex  system  like  the  present,  it  is 
probable  that  the  value  just  obtained  for  h  is  not  a  sufficiently 
close  approximation  to  answer  our  purpose.  We  will  therefore 
make  a  second  calculation,  using  7  as  a  second  approximate  value 
for  h. 

Carrying  the  calculation  through  precisely  as  above,  we  obtain 
the  following  values : 

h  =  7.32 
h'  =  16.42 
h"  =  29.60 
h'"  =50.00 

Calculating  the  various  discharges  under  these  piezometric 
heads,  calling  those  through  the  different  sections  of  48-in.  pipe 
commencing  at  the  lower  end,  Q,  Q,'  Q,"  Q,'"  and  those  through 
the  corresponding  24-in.  branches,  q,  Q,'  q,"  we  have: 

Q    =  122.05 

(/  =  14.30 

Q  +  q  =  136.35 

Q'  =  136.10 

q  =  27.62 

Q"  -f  q  =  163.72 

Q"  =  163.75 

q"  =  39.72 

Q"  -t-  q"  =  '203.47 

Q'"  =  203.75 

These  results  show  a  very  close  agreement. 


34  PRACTICAL    HYDRAULIC    FORMULAE. 

It  is  worthy  of  note  that  the  total  discharge  in  this  case  is  not 
greatly  increased  over  that  obtained  with  a  single  branch  situated 
500  feet  from  the  reservoir.  In  general  it  will  be  found,  as  in  these 
two  cases,  tnat  when  a  main  is  tapped  at  a  certain  point  by  a 
single  branch,  the  total  discharge  is  comparatively  slightly  in- 
creased by  the  introduction  of  a  series  of  similar  branches  placed 
below  the  first  junction.  The  position  of  the  first  branch,  however, 
has,  as  the  above  examples  show,  a  very  great  influence  both  on 
the  volume  of  discharge  and  the  form  of  the  hydraulic  grade 
line.  This  latter  feature  merits  careful  attention. 

It  will  be  interesting  to  study  the  effect  upon  the  flow  through 
such  a  system  as  we  have  been  just  considering,  when  the  condi- 
tions are  somewhat  changed.  For  instance,  in  the  last  example 
let  us  suppose  that  the  three  branch  pipes,  instead  of  having  each 
an  equal  rising  grade  of  4  feet  in  their  length  of  500  feet,  have  ris- 
ing grades  respectively  of  4  feet,  12  feet  and  24  feet  in  500,  commenc- 
ing at  the  lower  branch,  all  other  conditions  remaining  the  same. 

Assuming,  as  before,  an  approximate  value  for  h  of  9  feet,  we 
get,  as  before 

h'  =  20.52 

Our  next  equation  will  be  : 


i  / 

11.52  H y     17  04  =        y     h"-  20.52 

h"  -  35.H1 


Again : 


i        / 

A/    15.2'J    +   --     A/   23.62=      A/  h"'—  35,81 


This  value  is  sufficiently  near  the  given  one  of  50,  to  warrant 


PRACTICAL    HYDRAULIC    FORMULAE.  35 

our  using  it  to  obtain  pretty  close  approximate  values,  by  propor- 
tion, as  follows : 

h  -  8.00 
h'  =  18.24 
h"  -  31.&3 
h'"  =  50.00 


Whence  we  obtain  the  following  discharges 

Q  =  12   56  A/ -?*-=  127.6 
"       0-31 


=     3.14    A/-     -  =      15.7 
V     0  32 
Q  4-   q  =   1433 


0.31 


0.32        

Q'  X  q  =  164  0 


=      166.3 
0.31 


-  =       22.0 
0.32 


Q"  4.  q"   =    188.3 

4 
f 


Q'"  =  12.564        7'2'68       =     192.3 


0.31 

This  shows  a  pretty  fair  agreement  between  the  volumes  dis- 
charged, the  discrepancies  being  due  to  the  fact  that  our  assumed 
value  of  h  was  not  sufficiently  close  for  a  fine  calculation.  The 
figures  are  near  enough  however,  to  serve  the  purpose  of  showing 
to  how  small  an  extent,  comparatively,  the  results  are  changed  by 
the  very  considerable  changes  made  in  the  inclination  of  the 
branch  pipes.  Later  on  we  shall  have  occasion  to  notice  more 
fully  the  small  relative  changes  made  in  the  volumes  discharged 
through  given  pipes  by  changes  of  grade  ;  for  the  present  we  will 
only  call  attention  to  the  slight  variations  produced  in  the  hy- 
draulic grade  line,  as  determined  by  the  piezometric  heads. 


CHAPTER  III. 

Numerical  example  of  a  system  of  pipes  for  the  supply  of  a  town— Establishment  of 
additional  formuhe  for  facilitating  sucli  calculations— Determinations  of  dia- 
ameters— Pumping  and  reservoirs—  Caution  regarding  calculated  results- 
Useful  approximate  f or mu* 02— Table  of  5th  powers  -Preponderating  influence 
of  diameter  over  grade  illustrated  by  example.  —Maximum  velocities.  (Note.) 

As  a  farther  study  of  a  system  of  pipes  to  deliver  water,  let  us 
suppose  a  town  divided  by  intersecting  streets  into  blocks  1,000  ft. 
sq.,  as  shown  in  Fig.  9.  We  will  suppose  that  the  proposed  water 
supply  requires  a  total  volume  of  3  cu.  ft.  per  second,  equal  to  say 
800.000  U.  S.  gallu.  in  10  hours. 


-  j 


('7°) 


(<7± 


1^2. 


Fig.     9. 


PRACTICAL    HYDRAULIC    FORMULAE.  37 

The  water  is  to  be  introduced  by  a  central  main  A  B  C,  and 
delivered  east  and  west  by  the  side  mains  D  D',  E  JEf,  FFr,  G  G', 
and  H  H'.  At  the  extremities  of  these  mains,  the  water  is  to  be 
delivered  at  the  elevations  above  datum  indicated  by  the  figures 
placed  in  brackets.  The  side  mains  D  D',  and  E  E',  are  to  deliver 
each,  east  and  west,  \  cu.  ft.  per  second,  which  quantity  we  will 
suppose  is  to  bo  carried  through  the  whole  length  of  the  pipe,  and 
delivered  at  its  extremity,  without  regard  to  the  quantities  drawn 
off  en  route  by  the  service  pipes  and  smaller  north  and  south 
mains.  This  will  secure  a  good  delivery  of  water  in  case  of 
fires.  The  total  delivery  of  the  above  two  side  mains  will  there- 
fore be  1  cu.  ft.  per  second.  The  remaining  three  side  mains  F F', 
G  G',  and  HH'.  are  to  deliver,  similarly,  ^  cu.  ft.  per  second  at  each 
extremity,  making  2  cu.  ft.  for  the  three. 

These  being  the  data,  we  will  suppose  the  problem  to  be  the 
determining  of  the  respective  diameters  of  the  pipes,  and  the 
height  to  which  the  water  must  be  raised  in  a  supply  reservoir,  or 
standpipe,  situated  somewhere  to  the  north  of  the  town. 

The  problem  thus  stated  is  indeterminate  and  admits  of  an 
indefinite  number  of  solutions,  for  we  may  either  use  large  pipes 
and  low  elevations,  or  small  pipes  and  high  elevations.  Practi- 
cally, however,  there  are  limitations  to  this;  for  in  the  first 
place  we  shall  naturally  be  restricted  as  to  the  height  to  which 
it  would  be  possible  or  advisable  to  raise  the  water,  and  secondly, 
experience  show?  that  we  should  confine  ourselves  within  cer- 
tain limits  as  regards  the  velocity  of  the  water  in  the  pipes. 

Generally  speaking,  these  velocities  should  not  exceed  such 
as  would  be  produced  by  a  fall  of  from  4  to  8  ft.  per  thousand,  ac- 
cording to  the  size  of  the  pipe;  the  greater  fall  belonging  to  the 
smaller  diameter.  (See  note  at  end  of  chapter.) 

Before  commencing  the  calculations,  it  will  be  well  to  establish 
certain  additional  formulae,  derived  from  (4),  which  are  frequently 
of  considerable  utility. 


38  PRACTICAL    HYDRAULIC    FORMULAE. 

When  the  length  and  diameter  are  constant: 

Q'*H' 


When  the  head  and  diameter  are  constant  : 

Q'*     L 


When  the  head  and  length  are  constant  : 


2}~°      Q*  (J 

When  the  head  and  discharge  are  constant  : 
D'5_L'  C' 
~U*  ~  L  G 

When  the  length  and  discharge  are  constant  : 


Z>*      H'C 

These  relations  indicate  that,  other  things  being  equal,  the 
squares  of  the  discharges  vary  directly  as  the  heads  and  the  fifth 
powers  of  the  diameters,  and  inversely  as  the  lengths;  and  that, 
other  things  being  equal,  the  fifth  powers  of  the  diameters  vary 
directly  as  the  squares  of  the  discharges  and  the  lengths,  and  in- 
versely as  the  heads. 

As  these  relations  are  generally  used  for  approximations,  the 
coefficients  may  be  dropped,  and  the  equations  written  in  this 
form  : 


(7) 


(8) 


«-/*^ 


PRACTICAL    HYDRAULIC    FORMULAE. 

~ 


8 


"  =  rHr 


»-/ 

Other  combinations  can  be  made  from  these  relations.    Thus  : 

(12) 

Commencing  now  with  the  west  side  of  the  main  HH',  we 
have  £  cu.  ft.  to  be  delivered  at  an  elevation  of  (160)  above  datum. 
As  the  pipe  will  be  a  comparatively  small  one,  we  will  assume  a 
grade  of  y^,  which  will  give  a  rise  of  16  ft.  between  the  extremity 
and  the  main  junction,  and  requires  an  elevation  of  piezometric 
head,  at  this  junction,  of  (176),  as  shown  in  the  figure 

To  obtain  the  proper  diameter  of  pipe  for  this  grade  and  dis- 
charge, we  have,  using  (4),  and  assuming  C  =  0.00076  as  a  probable 
value: 

U)2X1000  <  0.00076 
1)6  =  8  X  0.61 

whence  D5  =  0  017304 
and  D  =  0.444. 

Or,  for  the  next  highest  even  inch : 
D  -  6  inches. 

As  regards  the  diameter  of  the  pipe  on  the  east  side,  since  the 
length  and  discharge  are  the  same  as  for  the  west  side,  and  only 
the  heads  vary,  being  respectively  16  and  36  ft ,  it  can  be  obtained 
by  means  of  (11). 

Thus: 


D-=  4    _L^_L^I 

V  36 

If  =  0.3777 

or,  for  next  highest  even  inch  : 

D'  =  5  inches. 


40  PRACTICAL    HYDRAULIC    FORMULAE. 

The  above  head  of  18  ft.  per  thousand  produces  a  velocity  of 
flow  in  a  5  in.  pipe  of  a  little  over  3  ft.  per  second,  which  is  some- 
what greater  than  it  should  be.  If  the  limit  of  velocity  is  over- 
stepped to  any  considerable  degree  in  a  system  of  pipes  such  as 
we  are  considering,  it  would  be  best  to  use  a  larger  pipe  and 
check  its  flow  dowja  to  the  desired  delivery  by  means  of  a  gate  or 
stop  cock  placed  near  its  upper  end,  the  effect  of  which  will  be  to 
diminish  the  head.  In  the  present  instance,  the  excess  of  velocity 
is  probably  not  sufficient  to  render  tliis  precaution  necessary. 

The  elevations  are  such  that  the  above  diameters  of  6  and  5 
ins.  are  also  proper  for  the  side  mains  G  G\  F  F. 

It  is  now  necessary  to  calculate  the  diameter  of  the  central 
main  from  B  to  C.  This  main  might  be  divided  into  two  parts, 
that  between  F  F'  and  G  G'  and  that  between  G  G'  and  H  H', 
but  we  will  calculate  it  on  the  supposition  of  a  uniform  diameter, 
capable  of  delivering  the  entire  volume  of  |  cu.  ft.  per  second  as  far 


Assuming  a  probable  value  of  G  =  0.00066,  we  have  from  (4)  : 


J6 
—  —  X  1.32 


6.1 

whence  :  y*  =  o.s«47 

and  :  D  =  0.826  =  10  ins. 

Taking  now  the  mains  E  E'  and  D  D',  and  beginning  on  the 
west  side,  assuming  as  before  a  grade  of  8  ft.  per  1,000,  we  find  the 
length  and  head  equal  to  those  of  F  F'  etc.  the  only  difference 
being  the  quantity  it  is  desired  to  deliver,  which  is  now  J  cu.  ft. 
as  against  $  in  F  F'.  1  he  relation  (9)  is  therefore  applicable,  and 
we  have : 


7,  -          /  ! 

V  ~   A  /    0.017304  X  — 

r  16 

i 

whence  :  j/«  ^  0.0097335 


PRACTICAL    HYDRAULIC    FORMULAE.  41 

and  D-  ^  0.395 

or,  say,  D'  =  5  ins. 

The  mains  on  the  east  side  are  determined  as  before  : 

5 

16 

"  ~  I/     0.00973*5    X    — 
31 

Z)'   =  0.346 

This  is  not  quite  4^  ins.  but  to  ensure  the  desired  delivery,  it 
will  be  best  to  take  the  next  highest  even  inch,  and  call  it  5  ins. 

As  regards  the  central  main  from  A  to  B,  we  find  two  grades, 
the  upper  one  y^  and  the  lower  !&ga.  The  lower  section  must 
deliver,  under  a  grade  of  T£g(j,  all  the  water  required  for  F  F',  G 
G\  and  H  H',  aggregating  2  cu.  ft.  per  second.  Using  (4),  and 
taking  0.00066  as  a  probable  value  of  C  we  have : 

nr,  =  *  X  0.66 
6.1 

whence :  z>~  =  o.  4028 

and  :  D  =  o.846 

This  is  very  nearly  10]  ins.,  and  a  10  in.,  pipe  would  answer, 
though  12  ins.  would  be  better. 

The  upper  section  must  deliver  2.5  cu.  ft.  per  second,  under  a 
grade  of  TT/fo.  Taking  the  same  probable  value  of  C,  we  have : 

^3    __  6.25  X  0.66 
2.05 

whence  :  D  =  1.237 

which  we  can  take  as  either  15  or  16  ins. 

This  diameter  might  have  been  obtained  from  that  of  the  lower 
section,  by  means  of  (12).  Thus : 

10        6.25 

D'°  =  0.4328    <    —  X   

5  4 

D'    =   1.287 

This  last  formula  might  have  been  used  throughout,  but  (4)  is 
abjut  as  short  and  convenient;  frequently  more  so. 


42  PRACTICAL    HYDRAULIC    FORMULAE. 

The  diameters  being  thus  determined,  the  quantities  should 
be  verified  by  (3).  They  will  be  found  somewhat  in  excess  of 
those  proposed,  owing  to  the  general  increase  of  the  diameters. 

As  regards  the  height  to  which  the  water  must  be  raised,  the 
data  show  that  3  cu.  ft.  per  second  must  be  raised  to  a  sufficient 
height  to  reach  D  D'  at  an  elevation  of  (201)  above  datum.  If  we 
adopt  a  grade  of  TTJ%^,  the  proper  diameter  of  the  pipe  would  be  : 

9X   0.65 

b*  = 

2.44 
D  =  1.32 

or: 

1)  =  16  ins. 

If,  instead  of  pumping,  the  water  were  collected  in  a  reservoir 
by  damming  up  the  natural  flow  of  some  stream,  and  the  dam 
were  of  necessity  situated  at  an  elevation  so  great  that  a  danger- 
ous pressure  is  apprehended,  it  would  be  necessary  to  first  receive 
the  water  into  a  distributing  reservoir  situated  at  a  lower  level,  or 
else,  as  a  less  advantageous  expedient,  to  reduce  the  pressure  by 
gates,  properly  located  for  the  purpose. 

It  should  be  well  understood  that  all  the  above  assumed  data, 
particularly  such  as  relate  to  heads,  are  subjected  to  considerable 
variation  in  actual  practice.  All  the  calculations  have  been  based, 
of  necessity,  upon  the  hypothesis  that  the  exact  allotted  volume 
per  second  is  being  simultaneously  drawn  from  the  whole  system. 
This  would  rarely  be  the  case;  for  at  any  given  second,  the 
draught  would  be  liable  to  fluctuate  greatly  from  the  average. 
Indeed,  these  calculations  should  only  be  regarded  as  fixing,  with 
some  degree  of  approximation,  the  proper  relative  discharges  ai^d 
pressures  at  the  different  points  supplied. 

The  remaining  north  and  south  pipes  should  be  calculated  in 
the  same  way.  Thus,  those  below  F  F'  on  the  west  side  discharge 
1-6  cubic  ft.  with  a  grade  of  VOS00.  This  would  require  a  4-in  pipe. 
The  draught  from  these  would  somewhat  lower  the  piezometric 


PRACTICAL    HYDRAULIC    FORMULAE.  43 

heads  at  their  junctions  with  the  side  mains.  Ill  a  fine  calcula- 
tion, these  reductions  should  be  worked  out,  as  was  done  in  the 
previous  example  of  branch  pipes ;  in  general,  however,  and  in 
cases  where  the  whole  supply  is  supposed  to  be  carried  through  to 
the  extremity  of  the  mains,  as  was  done  in  the  present  instance, 
and  where  a  liberal  interpretation  has  been  given  to  the  calcula- 
tion of  diameters,  this  is  not  indispensable.  At  the  same  time, 
it  should  be  a  guiding  principle  of  water-works  engineering,  that 
a  few  hours  spent  in  the  office,  in  what  may  sometimes  be  con- 
sidered an  over  refinement  of  calculation,  is  by  no  means  a  waste 
of  time,  and  frequently  enables  one  to  make  advantageous  and 
economical  modifications  in  a  project  of  distribution. 

It  may  here  be  noted  that  (12)  admits  of  being  put  into  a  very  con- 
venient form  for  rapid  approximations.  To  do  this,  we  have  only 
to  calculate  the  discharge  of  a  pipe  1  ft.  in  diameter,  with  a  fail  of 
1  ft.  per  thousand,  and  to  refer  all  other  discharges  with  the  fall 
per  thousand  feet  to  it,  in  order  to  obtain  the  corresponding  diam- 
eter. The  quantity  discharged  by  the  above  pipe  is  0.961  cu.  ft.  per 
second,  and  the  square  of  the  same  is  0.924.  Equation  (12)  may 
then  be  written : 


or  very  nearly  : 


D=|/    -  (13) 

we  have  also  very  nearly :   Q,  =  v  ^  x  H  iu) 

These  last  formulae,  it  will  be  perceived,  are  based  on  the  fact 
that,  given  a  certain  probable  degree  of  roughness,  a  pipe  1  ft.  in 
diameter,  with  a  fall  of  1  ft.  in  a  thousand,  will  deliver  1  cu.  ft.  of 
water  per  second.  If  we  desire  to  apply  them  to  smooth,  clean 


44  PRACTICAL    HYDRAULIC    FORMULAE. 

pipes,  we  have  oiily  to  halve  the  co  efficient  for  a  12-in.  pipe,  which 
will  be  equivalent  to  writing  the  above  formulas  thus : 

5 

/Q2 

D     {/    -  (15) 

>   Iff 


Q,=    V  D*  x  1H  (16) 

These  formulae  will  be  found  of  very  great  utility  in  arriving 
quickly  at  approximate  results.  They  can  be  advantageously  used 
in  sketching  out  a  network  of  pipes  such  as  we  have  just  been  con 
sidering.  To  facilitate  their  use  the  following  table  of  fifth  powers 
has  been  calculated.  This  table  indicates,  by  inspection,  the  diam- 
eters in  inches  corresponding  to  the  fifth  roots  of  the  right-hand 
side  of  the  equations,  expressed  in  feet. 


Diameters  in  inches. 

Fifth  Powers  in  feet. 

Diameters  in  ins. 

Fijth  Powers  in  feet 

3 

0.000977 

22 

20.72 

4 

0.004115                            24 

32.00 

5 

0.01256                              26 

47.75 

6 

0.03125 

28 

69.17 

8 

0.1317 

bO 

97.66 

10                                 0.4019 

32 

134.9 

12 

1.0000                               34 

182.6 

14 

2.1615                                36                                    243.0 

16 

4.214 

40 

411.5 

18 

7.594 

42 

525.2 

20 

12.86 

48 

1024.0 

All  the  diameters  which  have  been  already  calculated  can  be  ob- 
tained very  nearly  by  the  use  of  (13).  Relations  (13)  and  (14)  might 
also  have  been  used  in  some  of  the  previous  examples. 

Formulas  (13)  and  (14)  serve  to  show  the  comparatively  small  in- 
fluence of  grade  as  affecting  the  volumes  discharged,  which  point 
has  been  already  alluded  to,  and  the  preponderating  influence  of 
diameter.  Thus,  we  see  by  the  above  formulas,  that  for  a  diameter 
of  1  ft.  and  a  fall  of  1Ty\)0,  the  volume  of  discharge  is  1  cu.  ft.  If  we 
wish  to  double  this  discharge  by  increasing  the  fall,  we  must  adopt 
a  grade  of  ^oo,  i-  e.,  we  must  quadruple  the  fall.  If,  on  the  other 
hand,  we  wish  to  produce  the  same  result  by  increasing  the  diam- 


PRACTICAL  HYDRAULIC    FORMULAE.  45 

eter  without  changing  the  grade,  we  Deed  only  adopt  a  diameter  of 
1.32  ft.  and  even  a  little  less,  on  account  of  the  decrease  in  the  co- 
efficient. That  is  to  say,  to  double  the  discharge,  we  must  increase 
the  fall  300  per  cent.,  or  the  diameter  32  per  cent. 

NOTE.— In  completion  of  what  has  been  already  said  in  this  chapter  (p»ge 
37),  regarding  the  limit  of  velocities  for  pipes  01  different  diameters,  the  follow- 
ing table  (founded  upon  that  given  by  Mr.  Fanning)  indicates  pretty  closely  the 
maximum  velocities  which  it  is  generally  advisable  to  produce: 

Diameter  in  inches,  6          12          18          24         30          36          42          48 

Velocity  in  ft.  per  sec.,        2.5        3.5        4.5        5-5        6.5        7.5        8.5        9.5 


CHAPTER    IV. 

Use  of  formula  U  illustrated  l>\i  numerical  examvle  of  compound  system  com- 
bined with  branches— Comparison  of  results— Rough  and  smooth  pipes—  Hpes 
communicating  with  three  reservoirs— Numerical  examples  under  varying  con- 
ditions—Loss  of  head  from  other  causes  than  friction— Velocity,  entrance  and 
exit  heads— Numerical  examples  and  general  formulce— Downward  discharge 
through  a  vertical  pipe,— Other  minor  losses  of  head— Abrupt,  changes  of  dia- 
meter—Partially  opened  valve— Branches  and  bends—  Certrifugal  force— Small 
importance,  of  all  losses  of  head  except  frictional  in  the  case  of  long  pipes- 
All  suclt,  covered  by  "even  inches,1"  in  the  diameter. 

As  an  illustration  of  the  use  of  (14)  we  will  calculate  by  its  aid  the 
discharge  from  a  reservoir,  tapped  at  a  depth  of  50  ft.  by  a  horizon- 
tal compound  system  consisting  successively  of  2,000  ft.  of  12-in. 
pipe,  2,000  ft.  of  24-in.  pipe  and  2,000  ft.  of  12-in.  Each  of  these 
three  lengths  of  pipe  are  themselves  tapped  midway  by  a  6-in.  pipe, 
laid  horizontally,  the  one  nearest  the  reservoir  having  a  length  of 
3,000ft;  the  next,  1,000  ft.,  and  the  last,  500ft.  (See  Fig.  9,  bis) 
All  the  pipes  being  open,  it  is  desired  to  find  the  piezometric  heads 
/<,  h',  U",  h"f,  h"",  at  each  branch  and  change  of  diameter,  and  the 
volumes  discharged  by  each  branch  and  section  of  main  pipe. 

Beginning  at  the  lower  end  and  assuminging  6  ft.  as  an  approxi- 
mate value  of  h,  we  have  from  (14),  H  always  representing  the  fall 
per  1,000. 

V  ~6~4r  ^  =    V  k'  _G 
h'  =  15  36 


^936  =    *  32  (k"  —15.36) 
k"  =  15.65 


1)36  +     153i5  =     32—  ses 

h'"  =  16.09 


=  30.17 


*'  14  08  +  Vl%r  =   v'  h -  30.17 

h =  48.82 


PRACTICAL    HYDRAULIC    FORMULAE.  47 

Comparing  this  value  with  the  given  height  50,  we  may  increase 
all  the  preceding  values  of  h,  h',  etc.,  in  the  proportion  of  4i^-2.  But 
in  practice  we  would  not  wish  to  reckon  on  the  total  head,  and  it 
would  be  preferable  therefore  to  let  the  values  stand  as  they  are. 

We  will  now  calculate  the  quantities,  calling  those  discharged 
from  the  successive  sections  of  main  pipe,  beginning  at  the  lower 
end,  Q  Q',  Q",  Q"\  Q"",  and  Q""f,  and  those  discharged  by  the 
branches,  beginning  also  at  the  lower  end,  q,  q'y  q"  respectively, 
using  both  (3)  and  (14).  The  results  given  by  (14)  naturally  check 
exactly,  since  they  depend  directly  upon  the  method  used  in  deter- 
mining h,  h',  etc. 


By  (3) 

By  (14) 

Q 

=  2  39 

2.45 

Q 

=     .56 

.61 

Q 

_|_  q 

=  2.95 

3.06 

Q 

=  2/96 

3.06 

Q 

=  2.99 

3.05 

Q' 

=     .63 

70 

+  <!' 

=  3.64 

S.75 

8" 

'      =  3.68 
'  '   =  3.63 

3.75 
3.75 

Q" 

=     .52 

.f« 

'+«" 

=    4.15 

4.31 

V" 

=    4.18 

4.32 

The  above  example  was  very  favorable  to  the  use  of  (14),  because 
of  the  lengths  assumed  for  the  different  pipes,  but  in  almost  all 
cases  it  will  greatly  reduce  the  volume  of  calculation,  and  fre- 
quently give  sufficiently  close  results.  Indeed,  as  all  these  calcu- 
lations are  merely  approximations,  and  as  we  have  taken  our  co- 
efficients pretty  high,  it  would  no  doubt  often  be  found,  could  the 
actual  discharges  be  measured,  that  the  apparently  less  exact  for- 
mula gave  the  more  correct  results. 

In  all  the  previous  examples,  the  coefficients  for  rough  pipes 
have  been  used.  It  is  well  to  remember  that,  as  is  shown  by  (15) 
and  (16),  the  discharge  of  a  clean  pipe  of  given  diameter  is  about  41 
per  cent,  greater  than  that  of  a  rough  pipe  of  the  same  diameter ; 
also  that  the  diameter  of  a  clean  pipe  discharging  an  equal  volume 


48  PRACTICAL  HYDRAULIC    FORMULAE. 

with  a  rough  one,  will  be  about  88  per  cent,  of  the  latter.  Between 
these  limits  of  smoothness  and  roughness,  there  are,  of  course,  an 
indefinite  number  of  gradations. 

A  very  interesting  investigation  is  that  of  a  system  of  pipes 
communicating  with  two  reservoirs,  and  discharging  either  freely 
in  the  air,  or  into  a  third  reservoir  situated  at  a  lower  elevation  as 
shown  in  Fig.  10. 


Fig.  10. 

Let  us  suppose  the  water  surfaces  iu  A  and  B  to  be  respectively 
10o  and  80  ft.  above  the  water  surface  in  C,  and  that  all  the  pipes 
shown  in  the  figure  are  12  ins.  in  diameter.  Let  the  total  length  of 
pipe  -from  A  to  C  be  4,000  ft. 

If  communication  were  shut  off  from  B,  the  flow  would  be  di- 
rect from  A  to  C:  if  communication  were  shut  off  from  (7,  it  would 
be  direct  from  A  to  B.  If  A  were  shut  off,  the  flow  would  be  from 
B  to  C.  If  all  the  communications  were  wide  open,  we  desire  to 
know  whether  the  flow  would  be  from  A  to  B  and  C,  or  from  A  and 
B  to  C\  and  in  either  case,  to  know  the  piezometric  head  h,  at  the 
junction  D,  and  the  volumes  discharged. 

First,  let  the  junction  D  be  situated  midway  in  the  4,000-ft.  pipe 
joining  A  and  (7,  and  let  the  length  B  D  be  1,000  ft.  Let  us  for  a 
moment  revert  to  the  supposition  that  B  is  shut  off.  The  flow 
would  then  be  from  A  to  C.  the  hydraulic  grade  line  would  be  a 
straight  line  joining  the  surfaces  A  and  C,  and  under  our  present 


PRACTICAL    HYDRAULIC    FORMULAE.  49 

hypothesis  that  the  junction  D  is  in  the  middle  of  A  C,  the  piezo- 
metric  head  h  would  be  50  ft.  above  the  surface  of  the  lower  reser- 
voir C.  But  B  is  supposed  to  be  80  ft.  above  the  same,  and  therefore 
the  flow  must  be  from  A  and  B  to  C.  We  might  at  first  sight  sup- 
pose that  the  flow  from  B  to  C  would  be  in  virtue  of  the  head  80  — 
50  =  30  ft.,  which  is  the  difference  of  level  between  B  and  the  piezo- 
metric  head  at  the  junction ;  but  just  as  a  branch  drawing  water 
from  a  main  pipe  lowers  the  piezometric  head  at  the  junction,  so 
does  a  branch  discharging  into  the  main  pipe,  raise  it.  It  is  neces- 
sary to  see  what  the  height  h  will  be  in  the  present  case. 

The  quantity  discharged  into  C  is  equal  to  the  sum  of  the  quan- 
tities passing  from  A  and  B,  All  areas  and  coefficients  being 
equal,  and  all  reductions  made,  we  have  : 


I/- 

"        o 


whence : 


/  W 

h  =  65  -f  |/    4.000  —  90  h  H 


and,  by  successive  approximations  : 

It.  =  74 

Using  this  value  of  h  in  (3),  we  obtain  the  different  discharges 
as  follows : 

Q  =5.88 
Q'  =  3.48 
Q"  =  2.37 

This  gives  a  very  close  agreement  in  the  relation  Q  —  Q'  -f-  Q". 

Suppose  now  that  the  diameter  of  the  branch  B  D  be  reduced 
to  6  ins.  all  the  other  conditions  remaining  the  same.  Still  regard- 
ing the  co-efficients  as  equal,  in  order  to  get  rapidly  at  an  approxi- 
mation, factoring  the  areas  and  simplifying,  we  have: 


h 

\       o 


/..     h  .  ./._!_* 

2 


whence :  ie.5  //  ==  840  +  4  v  s.ooo  —  iso  k  +  /«» 


50  PRACTICAL    HYDRAULIC    FORMULAE, 

and,  by  successive  approximations: 

h  =  58 

This  value  of  h  gives  the  following  quantities  : 

?=  5.21 
-4.43 
'  =  1.08 

A  tolerably  close  check,  but  showing  that  the  true  value  of  h  is 
a  little  greater  than  the  even  58  ft.  at  which  we  have  placed  it. 

Let  us  now  suppose  that  the  pipe  B  D  is  increased  to  a  diam- 
eter of  36  ins.  all  the  other  conditions  remaining  as  before. 

Then- 


/  h  h          / 

A/    —  =  I/    50 h  9  4/    80  —  h 

2         *  2 

whence  :  />  =  79.90 

Giving:  Q  =  c.in 

Q'  =  3.065 
Q"  =  2.816 

a  close  approximation ;  the  true  value  of  h  lies  between  79.85  and 
79.90. 


As  /i  increases  with  the  diameter  of  the  pipe  B  D,  it  might  at 
first  seem  as  though,  by  indefinitely  increasing  the  diameter,  h 
might  be  so  increased  as  to  cause  a  flow  from  A  into  B.  A  mo- 
ment's reflection,  however,  will  show  that  under  the  assumed  con- 
ditions, the  diameter  can  never  be  sufficiently  increased  to  cause  a 
flow  towards  B.  For  it  has  been  seen  that  when  B  is  shut  off,  the 
piezometric  head  at  D  is  50  ft.  It  is  raised  by  opening  the  commu- 
nication with  B,  and  allowing  water  to  flow  into  the  main  from  B. 
It  is  evidently,  therefore,  an  essential  condition  of  the  increase  of 
piezometric  height  that  the  flow  should  be  from,  not  to,  the  reser- 
voir B. 

But  the  effect  will  be  different  if  the  junction  D  be  sufficiently 
advanced  towards  the  reservoir  A.  Let  us  suppose  the  positions  of 
the  three  reservoirs  to  remain  the  same,  all  the  pipe  diameters  to 
be  12  ins.,  and  the  point  of  junction  of  the  pipe  B  D  to  be  placed  at 
500  ft.  from  A  (Fig.  11).  If  communication  with  B  were  shut  off, 


PRACTICAL    HYDRAULIC    FORMULAE. 


51 


the  piezometric  height  at  D  would  be  87.5  ft.  There  would  there- 
fore be  a  flow  from  A  to  B  and  C  when  the  pipe  leading  to  B  was 
open.  But  this  flow  would  not  take  place  under  the  head  87.5,  for 
the  draught  towards  B  would  lower  it. 


Fig.    11. 

To  ascertain  the  true  value  of  h  at  the  point  D,  we  have  the 
relation : 

''Too  —  /< 


2,500 


simplifying 


loo-  h 


/   h 

-  v 


/h- 

+  I/— 


47  h  =  4060—  11.86   V  7i2  —  80  h 

whence,  by  successive  approximations : 

h  =  82.65  • 

Using  this  value  of  h  we  get : 

Q  =  5.695 
Q'  =  4.698 
Q"  =  .995 

When  B  is  shut  off,  in  the  above  system,  the  discharge  from  A 
to  Cis  4.83  cu.  ft.  per  second. 

In  all  that  precede?,  only  the  resistance  due  to  friction  has 
been  considered,  and  the  total  difference  of  level  between  the 
source  of  supply  and  the  discharge  has  been  taken  as  available 
for  overcoming  this  frictional  resistance.  In  the  case  of  long 


52  PRACTICAL    HYDRAULIC    FORMULAE. 

pipes,  where  the  velocity  is  comparatively  low,  this  resistance  is 
so  greatly  in  excess  of  all  the  others  that,  in  order  to  simplify 
calculations,  they  are  neglected.  This  leads  to  no  material  error 
in  cases  where  the  pipe  is  over  1,000  diameters  in  length. 

Attention,  however,  has  been  already  called  to  the  fact  that 
there  are  other  resistances  which  require  a  certain  proportion  of 
the  total  head  to  overcome  them,  leaving  only  the  remainder 
available  as  against  friction.  Indeed,  it  is  evident  if  we  assume 
all  the  head  to  be  consumed  by  frictional  resistance  alone,  the  wa- 
ter in  the  pipe  would  be  in  exact  equilibrium,  and  no  flow  could 
take  place. 

It  will  now  be  proper  to  show  how  the  total  loss  of  head,  from 
all  causes,  may  be  calculated.  And  first,  a  word  in  reference  to 
the  phrase  ''loss  of  head"  just  employed.  This  term,  often  met 
with  in  treatises  on  hydraulics,  may  occasionally  prove  confusing. 
It  is  really  little  more  than  a  convenient  abbreviation.  When  we 
speak,  for  instance,  of  "  the  loss  of  head  due  to  velocity,"  we  mean 
the  head,  or  fail,  theoretically  necessary  to  produce  that  velocity. 
Similarly,  when  we  speak  of  "the  loss  of  head  due  to  resistance  to 
entry,"  we  mean  the  amount  of  head,  or  pressure,  necessary  to 
force  the  fluid  vein  into  the  mouth  of  the  pipe  or  orifice,  against 
the  resistance  of  its  edges.  This  resistance,  it  may  be  remarked  in 
passing,  as  well  as  that  due  to  bends,  elbows,  and  branches,  shortly 
to  be  mentioned,  is  caused  by  the  fact  that  water  is  not  a  perfect 
fluid,  and  therefore  changes  of  direction  in  its  flow  require  a  cer- 
tain amount  of  force  to  break  or  distort  the  form  of  the  fluid  vein 
as,  though  to  a  very  much  less  degree,  would  be  the  case  with  a 
plastic  body  under  similar  circumstances.  The  property  of  water 
which  causes  these  resistances  is  called  its  viscosity. 

As  applied  to  long  pipes,  the  principal  "  loss  of  head,"  and  the 
only  one  hitherto  considered,  is  the  frialional.  The  term  thus  ap- 
plied means  the  height  or  pressure  necessary  to  overcome  the  fric- 
fion  of  the  water  passing  with  a  given  velocity  through  a  pipe  of 


PRACTICAL    HYDRAULIC     FORMULAE. 


53 


given  diameter.  Thus,  when  we  speak  of  the  frictional  loss  of 
head  per  1,000  ft.  in  reference  to  a  given  pipe,  we  mean  the  fall  per 
1,000  ft.  necessary  to  maintain  the  given  or  desired  velocity,  as 
against  friction. 

We  will  now  investigate  this  subject  by  means  of  the  following 
problem:  Two  reservoirs  (Fig.  12)  containing  still  water  and  hav- 
ing a  difference  of  level  of  30  ft.,  are  joined  by  a  pipe  12  ins.  in  diam- 
eter and  3000  ft.  long.  What  is  the  velocity  of  discharge  between 
the  upper  and  lower  reservoirs? 


Fig.   12. 

From  what  has  been  already  said,  it  will  be  seen  that  besides 
the  frictional  loss  of  head,  there  will  be  the  loss  of  head  due  to  ve- 
locity, and  that  due  10  entrance.  If  the  pipe  discharged  freely  in 
the  air  at  its  lower  end,  at  the  vertical  distance  of  30  ft.  below  the 
surface  of  the  water  in  the  upper  reservoir,  these  three  would  be 
the  only  losses  of  head  incurred,  and  their  sum  would  be  equal  to 
30  ft. ;  but  as  the  discharge  takes  place  in  a  reservoir,  the  surface 
of  the  water  in  which  is  supposed  to  cover  the  end  of  the  pipe,  to  a 
sufficient  depth  to  cause  the  discharge  to  take  place  in  still  water, 
there  is  the  further  loss  of  head  due  to  the  extinction  of  the  velocity 
which  is  dissipated  in  vortices.  This  loss  constituted  what  may  be 
called  the  backpressure  of  the  reservoir. 

In  solving  this  problem,  let  us  first,  as  heretofore,  neglect  all 


54  PRACTICAL  HYDRAULIC    FORMULAE. 

losses  except  frictional  ones.      We  have  then,  from  (1),  using  the 
above  data,  and  the  coefficient  for  rough  pipes : 

i 

=  0.00066  Fa 

100 

V2  =  15.15 

V  =  3  89  ft.  per  second 

The  head  theoretically  necessary  to  produce  this  velocity  is 

F* 
given  by  the  formula  derived  from  the  law  of  falling  bodies,  h 

2<7 
by  substitution  of  the  above  value  V  .    Thus  : 

15.15 

h= 

64.4 
h  =    0.2352 

Besides  this,  there  is^the  loss  of  head  due  to  entrance.  We 
have  already  seen  that  this  is  always  equal  to  about  half  the  velo- 
city head.  We  have  then  : 

ft 

ft  H =  0.3528 

2 

The  loss  of  head  from  back  pressure  of  the  water  in  the  lower 
reservoir,  being  that  necessary  to  extinguish  the  velocity  must  be 
equal  to  that  necessary  to  produce  the  same.  We  have  therefore 
for  the  total  losses,  outside  of  friction  : 

h 

h+ h  ft  =  0.588 

2 

And  the  head  available  for  overcoming  friction  becomes 

30  —  0.588  =  29.412 

We  must  now  recast  our  original  calculation,  using  29.4  ft.  in- 
stead of  30,  as  available  frictional  head.  Thus ; 

29.4 

=  0.00066  F2 

3000 

F2  =  14.8 

F  =  3.85 

This  is  a  very  small  reduction  from  the  velocity  already  ob- 


PRACTICAL    HYDRAULIC    FORMULAE.  55 

tained.  But,  in  order  to  see  how  our  previous  solution  is  affected 
by  the  change,  we  will  work  on  new  values  for  the  sub-heads. 
Thus: 

14.8 

h  = 

64.4 

h   =  0-23 
ft 

h  -\ \-h  =  0.575 

•2 
30—0.575  =  29.425 

leaving  the  previous  value  practically  unchanged. 

Let  us  now  see,  by  means  of  a  general  formula,  what  is  the 
amount  of  error  which  we  commit  when  we  ignore  all  resistances 
except  friction. 

Calling  F  the  actual  mean  velocity,  that  is  the  actual  volume 
discharged  divided  by  the  area  of  the  pipe  (3),  we  have,  in  the  case 
of  discharge  betweed  two  reservoirs,  as  shown  in  Fig.  12,  the 
following  subheads,  which  together  make  up  the  total  head  H: 

F2      F2      F2      L  C  F2 

a  =  -4-   -  +  -+  - 

20        40        20  D 

5F2       LC  F2 


40  D 

H  =  0.039  F2  +  L  C  V- 


D 

That  is  to  say,  by  using  (3)  which  gives, 
H  =  L  G  F2 


D 

we  make  the  error  of  omitting  a  distance  not  quite  equal  to  4  per 
cent,  of  the  square  of  the  velocity. 

In  long  pipes  this  is  a  very  trifling  amount. 

If  the  pipe  discharged  in  free  air,  we  would  have  : 

F2    v      L  C  F2 

//  =-+-+- 

20        40  D 

H  =   0  0-233  F2    +  L  C  Vs 
1) 


56  PRACTICAL    HYDRAULIC    FORMULAE. 

In  this  case  we  make  the  still  smaller  error  of  omitting  2£  % 
of  V*. 

In  all  eases,  having  obtained  V^  by  means  of  (1),  we  can  easily 
judge  from  the  nature  of  the  problem  whether  it  is  necessary  to 
take  account  of  these  errors.  In  designing  a  system  of  pipes, 
where  the  problem  generally  is  to  find  the  proper  diameter  for  a 
certain  discharge,  the  practice  of  taking  the  Dext  highest  even  inch 
will  almost  always  amply  suffice  to  cover  all  omissions. 

As  has  been  already  stated,  in  all  ordinary  circumstances  of 
pipe  laying,  the  horizontal  measurement  of  the  pipe  is  taken  in- 
stead of  its  actual  length.  It  is  only  in  special  cases  that  this  can- 
not be  done  The  extreme  limit  occurs  in  the  case  of  a  vertical 
pipe  discharging  from  the  bottom  of  a  reservoir.  This  constitutes 
a  very  interesting  special  case,  for  should  the  reservoir  be  of  indefi- 
nitely large  area  but  of  relatively  shallow  depth,  the  relation 

H 

—  tends  towards  unity  as  L.  and  consequently  H  increases.    The 

L 

velocity,  as  determined  by  (1)  tends  therefore  toward  : 


and  remains  constant,  no  matter  how  greatly  L  may  be  increased. 
If  we  apply  this  formula  to  a  12-in.  pipe  of  indefinite  length,  using 
the  coefficient  for  rough  pipes,  we  get, 


This  is  the  maximum  velocity  of  discharge  in  feet  per  second 
for  a  vertical  12-in.  pipe  under  the  given  circumstances. 

There  are  several  minor  losses  of  head,  besides  those  already 
considered,  which  are  liable  to  occur  from  changes  of  diameter, 
branches,  and  bends  or  elbows.  Our  experimental  knowledge  of 
the  effects  of  these  features  is  very  limited,  and  it  is  probable  that 


PRACTICAL    HYDRAULIC    FORMULAE.  57 

much  weight  should  not  be  attached  to  the  formulae  given  for  their 
determination.  A  brief  space  will  be  devoted  to  their  considera- 
tion, more  with  a  view  to  make  the  present  paper  complete  than 
for  any  practical  value  which  they  possess. 

When  water  passes  through  a  pipe  of  which  the  diameter  is  ab- 
ruptly changed,  at  a  certain  point,  to  a  greater  or  a  smaller  one, 
there  is  a  loss  of  head  due  to  the  eddies  formed  and  the  sudden 
contraction  of  the  fluid  vein.  In  practice  such  pipes  are  always 
joined  by  a  reducer,  or  special  casting,  which  forms  a  tapering  con- 
nection between  the  two.  This  greatly  diminishes  the  agitation  of 
the  water  in  passing  from  one  pipe  to  the  other.  It  would  seem 
however,  that  the  mere  change  of  velocity,  independent  of  such 
agitation,  causes  some  slight  modification  of  the  profile  of  the  hy- 
draulic grade  line:  and  it  will  be  well,  in  any  event,  to  give  for- 
mulae for  the  different  cases  which  may  occur  when  abrupt  changes 
take  place,  as  these  give  rise  to  the  maximum  retardation.  The 
following  formulae  are  taken  from  Claudel's  Aide  Memoire,  ninth 
edition. 

First.— When  the  change  is  from  one  pipe  to  another  of  smaller 
diameter,  we  have : 

y2 

//  =  0.49 

20 

whence  :  /<  =  O.OOOTG  V- 

V  being  the  velocity  of  the  water  in  the  smaller  pipe.  We  have 
seen,  by  examples  previously  given,  how  thU  velocity  may  be  ob- 
tained. 


Fig.  13 
Second.— If  the  water  (Fig.  13),  in  its  passage  from  the  greater 


58 


PRACTICAL    HYDRAULIC    FORMULAE. 


to  the  smaller  pipe,  passes  through  an  opening  in  a  thin  diaphragm, 
as  in  the  case  of  a  partially  opened  stop-cook,  we  have : 


V2  f     8          \2 
=  — •  ( l   I 

2  Q    \0.62  S'  / 


20 

in  which  V  is  the  velocity  in  B,  S  the  area  of  cross-section  of  B, 
and  S',  the  area  of  the  opening  in  the  diaphragm. 

Third. — When  the  flow  is  from  one  pipe  to  another  of  larger 
diameter : 

_(v-  vr- 

20 

in  which  V  =  velocity  in  small  pipe,  and  V  =  velocity  in  larger 
one.  When  the  water  passes  from  a  pipe  into  a  reservoir,  as  in  the 
case  lately  considered,  V  becomes  zero,  and  we  have,  as  already 
established  in  that  case : 

F2 


c, 

Fig.  14 


Another  loss  of  head  is  that  due  to  branches  (Fig.  14).  In  this 
case  the  water  flowing  from  A  with  a  velocity  F,  is  split  at  the 
junction,  part  passing  on  towards  B,  with  a  reduced  velocity  F', 
and  part  entering  the  branch  and  flowing  towards  C,  with  the  ve- 
locity V".  The  loss  of  head  occasioned  by  perturbations  of  the  wa- 
ter at  the  junction  has  not  been  satisfactorily  investigated.  When 


PRACTICAL  HYDRAULIC    FORMULAE.  59 

the  branch  leaves  the  main  at  a  right  angle,  this  loss,  as  deter- 
mined by  a  few  incomplete  experiments,  is : 

3  F'2 

h=  

20 

V"  being  the  velocity  in  the  branch.     We  have  already  seen  how 
this  velocity  may  be  calculated. 

If,  as  is  generally  the  case  in  practice,  the  branch  is  deflected 
gradually  instead  of  forming  an  abrupt  angle  of  90°,  the  vortices 
are  nearly  annulled,  and  the  only  loss  can  be  from  the  difference 
of  the  velocities  in  the  three  pipes.  Thus  for  B  and  C  respectively, 
we  have : 

F- 


For  bends,  or  elbows,  Navier's  formula  for  loss  of  head  is  : 

F*  /  \  A 

h   = I  0.0128  +  0.0186  R  I  — 

2flf  V  '  R 

in  which  V  =  velocity  of  flow,  E  =  the  radius  of  the  bend,  taken 
along  the  axis  of  the  pipe,  and  A  =  the  length  of  the  bend,  also 
measured  along  the  axis. 

It  will  readilv  oe  seen  how  very  trifling  the  loss  of  head  from 
this  cause  will  be  in  all  ordinary  cases. 

The  water  passing  around  a  bend  exercises  a  radial  thrust  upon 
it  which  may  sometimes  be  so  considerable  as  to  require  bracing 
against.  The  expression  'or  the  centrifugal  force  Fis : 

M  F2 

R 

in  which  M=  the  mass  of  the  liquid  in  motion,  V  =  its  velocity , 
and  R  =  the  radius  of  the  bend  measured  on  its  axis. 


60 


PRACTICAL    HYDRAULIC    FORMULAE. 


As  an  illustration,  we  will  suppose  a  pipe  24  ins.  in  diameter, 
through  which  the  water  flows  with  the  velocity  of  8  ft.  per  second, 
around  a  bend  of  8  ft.  radius. 

The  mass  of  the  liquid  in  motion  is  its  weight  divided  by  g. 
The  centrifugal  force,  therefore,  per  running  foot  is  : 

3.14  X  62.5      82 

F  = X  — 

32.2  8 

F  =  48.72  Ibs. 

If  the  bend  turns  a  quarter  circumference,  its  development  on 
the  axis  will  be  12.57  ft.,  and  the  total  thrust  on  the  bend  will  be 
48.72  X  12.57  =  612.4  Ibs. 

This  would  be  liable  to  be  intensified  by  sudden  changes  in  ve- 
locity, and  if  the  bend  is  not  well  abutted,  might  tend  to  draw  the 
joints. 


JB'ig.    15. 

Fig.  15  shows  the  manner  in  which  such  losses  of  head  as  we 
have  been  just  considering,  modify  the  profile  of  the  hydraulic 
grade  line.  The  dotted  line  shows  the  grade  as  determined  by  the 
calculations  which  we  have  already  made  for  a  line  of  pipes  of 
varying  diameter.  The  full  line,  broken  at  the  reservoir  and  at 
each  change  of  diameter,  shows  the  hydraulic  grade  as  modified  by 


PRACTICAL    HYDRAULIC    FORMULAE.  61 

losses  of  head  due  to  velocity  and  changes  of  diameter.  It  will  be 
understood,  of  course,  that  this  is  a  mere  random  sketch,  without 
reference  to  proportion. 

The  result  of  what  precedes  in  reference  to  all  losses  of  head 
other  than  friction,  shows  that  in  practice,  and  in  the  case  of  long 
pipes,  such  losses  exercise  but  a  trifling  influence.  A  very  small 
increase  in  the  diameter  of  the  pipe  over  that  obtained  by  calcula- 
tion based  on  frictlonal  head  alone,  such  as  would  naturally  be 
made  to  get  even  inches,  will  in  almost  all  cases  largely  cover  all 
losses  due  to  velocity,  entrance,  branches,  bends,  etc. 

THE    END. 


ENGINEERING  NEWS  is  a  weekly  journal  of  60  pages  10£  by  14  ins. 
in  size;  it  publishes  each  week  more  than  two  hundred  items  of 
news  relating  to  railroads,  water-works  and  miscellaneous  con- 
tracting intelligence;  it  is  especially  a  newspaper  for  engineers, 
railway  officials  and  contractors,  and  as  such  is  without  a  rival  in 
this  country. 

Officials  of  city,  town,  village,  township,  drainage  district,  or  any 
other  municipal  organization  will  find  it  the  CHEAPEST  and  MOST 
PROFITABLE  medium  for  advertising  their  public  works ;  city  engi- 
neers ;  railroad  engineers  and  managers  will  make  money  by  using 
its  columns  for  advertising  proposals;  civil  engineers  and  con- 
tractors will  find  it  to  their  advantage  to  have  in  it  a  permanent 
card,  giving  their  address  and  specialty,  and  everybody  who  has 
any  interest  in  public,  works  promotion,  planning  or  construction, 
will  find  it  a  good  investment  to  be  on  its  subscription  list. 

The  price  of  subscription  is  $5  per  year,  and  the  price  for  ad- 
vertising proposals  for  contracts  is  2O  Cts.  per  line,  which  is 
half  the  rate  of  the  city  dailies,  while  it  is  many  times  superior,  as 
it  goes  directly  to  the  class  for  which  it  is  intended. 

We  especially  solicit  therefore,  a  trial  of  our  advantages  in  this 
line  of  advertising.  IT  WILL  PAY  ADVERTISERS  EVERY  TIME,  and  it 
costs  very  little  when  its  efficiency  is  well  considered. 

We    hope  that    every  city    and    railroad  engineer  will  take  the 
trouble  to  recommend  this  journal  to  the  officials  who  control  the 
advertising  of  work  to  be  done.    They  can  do  such  in  good  faith, 
as  they  cannot  fail  to  appreciate  the  advantages  we  offer. 
Address  all  communications  to  : 

Engineering  News  Publishing  Co. 

TRIBUNE  BUILDING, 

NEW  YOKE  CITY. 


I    Vol.,  8vo.  644  pp  ,  2OO  Illustrations,  Cloth,  $5 

A  PRACTICAL  TREATISE 


ON 


Water-Supply  Engineering: 

RELATING  TO  THE 

HYDROLOGY,  HYDRODYNAMICS,  AND  PRACTICAL  CONSTRUC- 
TION OF  WATER- WORKS,  IN  NORTH  AMERICA, 

WITH  NUMEROUS 

Tables  and  Ilhistratiniis, 

BY  J.  T.  FANNING,  C.  E., 
Member  of  the  American  Society  of  Civil  Engineers. 


7th    Edition,  Revised,    Enlarged  and  New  Tables  and  Illus- 
trations   added. 


OOJNi  TEJIsT  J?S, 
SECTION  I.— Collection  and  Storage  of  Water,  and  its  Impurities. 

CHAPTEB  I.— Introductory.  CHAP.  II.— Quantity  of  Water  Required.  CHAP. 
III.  Rainfall.  CHAP.  IV.— Flow  of  Sti earns.  CHAP.  V.— Storage  and  Evapora- 
tion of  Water.  CHAP.  VI.— Supplying  Capacity  of  Watersheds.  CHAP.  VII.  — 
Springs  and  Wells,  CHAP.  VIIL— Impurities  of  Water.  CHAP.  IX.— Well,  Spring. 
Lake,  and  River  Supplies. 

SECTION  II.— Flow  of  Water  through  Sluices,  Pipes  and  Channels. 

CHAPTEB  X.— Weight,  Pressure,  and  Motion  of  Water.—  CHAP.  XI.— Flow  of 
Water  through  Orifices.  XIL— Flow  of  Water  through  Short  Tubes.  XIII.  - 
Flow  of  Water  through  Pipes  under  Pressure.  CHAP.  XIV.— Measures  of  Weirs 
and  Weir  Gauging,  CHAP.  XV.— Flow  of  Water  in  Open  Channels. 

SECTION  III.— Practical  Construction  of  Water-Works. 

CHAPTEB  XVI.— Reservoir  Embankments  and  Chambers.  CHAP.  XVII.— Open 
Canals.  CHAP.  XVIIL-  Waste  Weirs.  CHAP.  XlX.-Partitions  and  Retaining 
Walls.  CHAP.  XX.— Masonry  Conduits.  CHAP.  XXI.- Mains  and  Distribution 
Pipes.  CHAP.  XXII.— Distribution  Systems,  and  Appendages.  CHAP.— XXIII. — 
Clarification  of  Water,  XXIV.— Pumping  of  Water.  CHAP,  XXV.— Tank  Stand 
Pipes.  CHAP.  XXVI.— Systems  of  Water  Supply, 

APPENDIX.— Miscellaneous  Memoranda. 

ENGINEERING  NEWS  PUBLISHING  CO., 

TRIBUNE  BUILDING,  NEW  YORK. 


A  Civil  Engineer's  Library. 

Baker's'Engineering  Instruments .50 

Bouaearen's  Specifications  for  Iron  and  Steel  Highway  Bridges 25 

Bouscaren's  Speci fixations  for  Iron  and  Steel  Bailroad  Bridges 25 

Burr's  Bridge  and  Roof  Trusse~ 3.50 

Cooper's  Specifications  for  Iron  and  Steel  Highway  Bridges 25 

Cooner's  Specifications  tor  Iron  and  St^el  EaiLroad  Bridges 25 

Corfleld's  Treatment  and  Utilization  of  Sewage " 4.50 

Grand  all' a  Tables  of  Excavation  and  Embankment l.oo 

Cro«s'  Engineer's  Field  Book 25 

DuB  >is'  Strains  in  Framed  Structures  10.00 

Dunham's  Plat  and  Profile  Book,  20  miles — 1.00 

50  miles  2.00 

Ellis' Fire  Streams — 1.50 

Tanning's  Water  Supply  Engineering 5.00 

Flynn's  Hyd raulic  Taoles 50 

Flynu's  Flow  of  Water  in  Open  Channels ...'• 50 

Gilmore's  Limes,  Hydraulic  Cements  and  Mortnr —  —      4.00 

Gould's  Practical  Hydraulic  Formulae l.oo 

Gould's  Specifications  for  Dams  and  Reservoirs 25 

Greene's  Bridge  Trusses,Graphical  Method 2.50 

Greene's  Archie. 2.50 

Greene's  Boot' Trusses,  1.25 

Haswell's  Engineers' and  Mechanics' Pocket  Book 4.00 

Haupt's  Topographer 4.00 

Henck's  Field  Book  for  Railroad  Engineers 3.50 

Hering's  Bearing  Piles,  2d  Ed .20 

Hudson's  Tables  of  Excavation  and  Embankment 1-00 

Jackson's  Canal  and  Culvert  Table* 10.00 

Jackson's  Hydraulic  Manual 6.00 

Jacobs' Storage  Reservoirs  (new  edition) 50 

Johnson's  Theory  and  Practice  of  Surveying: 3.50 

Latham's  Sanitary  Engineering  (English  edition) 12.00 

Merriman's  Method  of  Least  Squares ....        250 

Osborn's  Tables  of  Moments  of  Inertia  and  Squares  of  Radii  of  Gyration. .      1.00 

Parsons'  Track,  A  Manual  of  Maintenance  of  Way .      2.00 

Parsons'  Turnouts  1.00 

Reed's  PbotogTaohy  aoplied  to  Surveying — 2.50 

Searl es'  Fiel d  Engineering 3,oo 

Shunk'sField   Engineer 250 

Smith's  Cable  Railways 2.50 

Staley  &  Pierson's  Separate  Svst«m  of  Sewerage  2,50 

Trautwine's  Civil  Engineers'  Pocket.  Bock 5.00 

Trautwine's  Field  Practice  of  Laying  out  Circular  Curves 2.50 

Trautwine's  Measurement  and  Cost  of  Earthwork 2.00 

Waddell's  Designing  of  Ordinary  Highway  Bridges    400 

Wellington's  Economic  Theorv  of  the  Location  of  Railways 5.00 

Wellington's  Computation  by  Diagrams  of  Excavation  and  Embankment.      4.00 

Wbipple's  Bridge  Construction 4.00 

No.  1,  Katte's  Specifications  for  Ccnstruction  of  Graduation  and  Masonry, 
Contents:  Articles  of  Agreement,  Formation,  Graduation,  Masonrv, 
(including  Specifications  for  Laying  Masonry  in  Freezing  Weather), 
Foundations,  Timber,  Iron  Work.  General  provisions  applicable 
to  all  work.  Indemnity  Bond.  16  pages.  Price  25  cts. 

No-  2.  Specifications  for  StanDard  Pile  and  Timber  Trestle  Bridging.  Price  5  cts. 
No.  3.  Specificat  ons  for  Cross  Ties.    Price  5  ct°. 
No,  4.  Specifications  for  Tra^k  Laying.     Price  10  cts. 

Any  book  will  be  sent,  postpaid,  on  receipt  of  the  price  by  the 

ENGINEERING  NEWS  PUB.  CO.,    TRIBUNE  BUILDIVG,  NEW  YOBK. 


